An Elementary Course in the Integral Calculus

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AN ELEMENTARY COURSE

IN

INTEGRAL CALCULUS

BT

DANIEL ALEXANDER MURRAY, Ph.D.

INSTRUCTOR IN MATHEMATICS IN CORNELL UNIVERSITY

FORMERLY SCHOLAR AND FELLOW OF JOHNS HOPKINS UNIYBRSITY

AUTHOR OF"

INTRODUCTORY COURSE IN

DIFFERENTIAL EQUATIONS''

"o"9{o"-

NEW YORK.;. CINCINNATI.:. CHICAGO

AMERICAN BOOK COMPANY



'/; r."":""."

' ^ i'

(viMV 21 189-;

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COPTRIOHT, 1898, BT

AMERICAN BOOK COMPANY.

JlirK"AY*8 INTBO. OAUX

W. P. 2



PREFACE

This book has been written primarily for use at Cornell

University and similar institutions. In this university the

classes in calculus are composed mainly of students in engi-eering

for whom an elementary course in the Integral Calculus

is prescribed for the third term of the first year. Their pur-ose

in taking the course is to acquire facility in performing

easy integrations and the power of making the simple applica-ions

which arise in practical work. While the requirements

of this special class of students have been kept in mind, care

has also been taken to make the book suitable for any one be-innin

the study of this branch of mathematics. The volume

contains little more than can be mastered by a student of

average ability in a few months, and an effort has been made

to present the subject-matter,hichis of an elementary char-cter,

in a simple manner.

Theobjectof the first two chapters is to give the student

a clear idea of what the Integral Calculus is, and of the uses

to which it may be applied. As this introduction is somewhat

longer than is usual in elementary works on the calculus, some

teachers may, perhaps, prefer to postpone the reading of sev-eral

of the articles until the student has had a certain amoxmt

of practice in.the processes of integration. It is believed, how-ver,

that a careful study of Chapters I.,II., will arouse the stu-ent's

interest and quicken his understanding of the subject.There may be some difference of opinion also as to whether



VI PREFACE

the beginner should be introduced to the subjectthrough

Chapter I. or through Chapter II. The decision of this ques-ion

will depend upon the point of view of the individual teacher.

So far as the remaining portion of the book is concerned it is

a matter of indifference which of these chapters is taken first;

and, with slight modifications, they can be interchanged. In

Chapter III. the fundamental rules and methods of integration

are explained. Since it has been deemed advisable to intro-uce

practical applications as early as possible, Chapter IV. is

devoted to the determination of plane areas and of volumes of

solids of revolution. Thesubjectf

Integral Curves, which is of

especial importance to the engineer, is treated in Chapter XII.

Many of the examples are original. Others, especially some

of those given in the practical applications, by reason of their

nature and importance, are common to all elementary courses

on calculus. In several instances, examples of particular interest

have been drawn from other works.

A list of lessons suggested for a short course of eleven or

twelve weeks is given on page viii. This list has been arranged

so that four lessons and a review will be a week's work.

It is hardly possible to name all the sources from which the

writer of an elementary work may have obtained suggestions

and ideas. I am especially conscious, however, of my indebted-ess

to the treatises of De Morgan, Williamson, Edwards,

Stegemann and Kiepert, and Lamb.

To my colleagues in the department of mathematics at

Cornell University, I am under obligations for many valuable

criticisms and suggestions. Both the arrangement and the

contents have been influenced in a large measure by our con-ferences

and discussions. As originally projected,he volume

was to have been written in collaboration with Dr. Hutchinson,

but circumstances prevented the carrying-out of this plan.



PREFACE vii

Chapters V., VI., in part, and Articles 2S, 73, in their entirety,

have been contributed by him. My colleagues have aided me

also in correcting the proofs.

Erom Professor I. P. Church of the College of Civil Engi-eering

and Professor W. Y, Durand of the Sibley College of

Mechanical Engineering, I have received valuable suggestions

for making the book useful to engineering students. Pro-essor

Durand kindly placed at my disposal, with other notes,

his article on ^'Integral Curves" in the Sibley Journal of

Engineering, Vol. XI., No. 4 ; and Chapter XII. is,with slight

changes, a reproduction of that article. I take this oppor-unity

of thanking Mr. A. T. Bruegel, Instructor in the kine-atics

of machinery, and Mr. Murray Macneill, Fellow in

mathematics in this university, the former for the interest

and care taken by him in drawing the figures, the latter for

his assistance in verifying examples and reading proof sheets.

D. A. MURRAY.

Cornell Univbbsitt.



LIST OF LESSONS SUGGESTED FOR A SHORT COURSE

[Rbvikws to follow Every Foueth Lesson]

10.

IL

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

Arts. 1, 2, 3.

Arts. 4, 5, 6, 7.

Arts. 8, 9, 10, 11.

Arts. 12, 17, 18, to Ex. 9.

Arts. 13, 18, Exs. 10-12, 19.

Arts. 14, 20.

Arts. 15, 16, 21.

Arts. 22, 23, Exs. 1-23, odd

examples.

Art. 23, Exs. 24-32, 24, 25,

Exs. 1-8, page 55.

Pages 56, 57, even or odd exam-ples,

Exs. 9-41, Exs. 42-47.

Arts. 26, 27, Exs. 1, 2, page 68.

Arts. 28, 29, Exs. 3, 4, page 68.

Exs. 5-14, page 68.

Art. 30.

Arts. 31, 32.

Page 76, Exs. 1-15.

Pages 76, 77, Exs. 16-26.

Arts. 33, 34.

Arts. 35, 36. Selected examples.

Arts. 37, 38, 39, 40.

Arts. 42, 43.

Art. 45.

Selected examples, pages 98, 99.

Arts. 46, 47, 48 (a).

Arts. 48 (rf),9, 50 (a),(c).Art. 51 (a).

Arts. 52, 53.-

Arts. 58, 59, 60.

Arts. 61, 62..

Arts. 64, 65, 66.

Arts.67, 68, 69.

Arts. 63, 70. Selected exam-ples,

pages 164, 165.

Art. 71. Selected examples,

page 165.


page 166.

Art. 73.


pages 164, 165.

Art. 75, Ex. 9, page 164.

Art. 76. Selected examples.

Art. 77, Ex. 1.

Selected examples, pages 162-

166.

Arts. 78, 79. Selected examples.

Art. 80. Selected examples.

Arts. 81, 82, 84, 85.

Arts. 86, 87, 88.

Arts. 89, 91, 92, 94, 95.



CONTENTS

CHAPTER I

Intbobation a Process of Summation

Abt. Page

1. Uses of the integral calculus. Definition and sign of integration.

1

2. Illustrations of the summation of infinitesimals....

2

3. Geometrical principle .7

4. Fundamental theorem. Definite integral 8

6. Supplement to Art. 3 13

6. Geometrical representation of an integral 14

7. Properties of definite integrals 16

CHAPTER II

Integration THE Inverse op Differentiation

8. Integration the inverse of differentiation 18

0. Indefinite integral. Constant of integration 22

10. Geometrical meaning of the arbitrary constant of integration.

23

11. Relation between the indefinite and the definite integral. .

24

12. Examples that involve anti-differentials 25

13. Another derivation of the integration formula for an area. .

27

14. A new meaning of y in the curve whose equation is y =/(x).De-ived

curves 29

15. Integral curves 33

16. Summary.35

CHAPTER ra

Fundamental Rules and Methods of Integration

18. Fundamental integrals '

. .

.3619. Two universal formulae of integration 39

20. Integration aided by a change of the independent variable . .41

21. Integration by parts"

. 44

22. Additional standard forms 47

23. Derivation of the additional standard forms. . . . ,

48

ix



X CONTENTS

Abt. Paok

24. Integration of a total differential 52

25. Summary 54

Examples on Chapter 111.55

CHAPTER IV

Geometrical Applications of the Calculus

26. Applications of the calculus . . . ... . .

.58

27. Areas of curves, rectangular coordinates 58

28. Precautions to be taken in finding areas by integration...

63

29. Precautions to be taken in evaluating definite integrals...

67

30. Volumes of solids of revolution 69

31. On the graphical representation of a definite integral...

73

32. Derivation of the equations of certain curves . .

*

. .

.74

Examples on Chapter IV 76

CHAPTER V

Rational Fractions

34. Case I 79

35. Case II 81

36. Casern 82

CHAPTER VI

Irrational Functions

38. The reciprocal substitution 84

39. Trigonometric substitutions 85

40. Expressions containing fractional powers of a-\-

hx only . .85

41. Functions of the form / " x^, (a + 6"^)"

\'Xdx,'mwhich w, 7i, are

integers 87

42. Functions of the form F(x, Vx^ + ax + 6) dx, F(u, v)being a

rational function of u,v 88

43. Functions of the form /(.r,V" x^-f

ax + 6)dx, f{u, v)being a

rational function of u, v 89

44. Particular functions involving Vax^ + dx-f-

c 91

45. Integration of x"'(a+ hx'^ydx : (a) by the method of undetermined

coefficients ; (6)by means of reduction formulae.

-

.

.93Examples on Chapter VI 98



CONTENTS xi

CHAPTER Vn

Iktboration of Trigonometric and Exponential Functions

Abt. Page

46. rsiu^xdic,cos^xdx,being an integer 100

47. Algebraic transformations 103

48. fsec*a;"fe,

fcosec"xdx 103

49. ftan"X(te,fcot"xdx 106

60. fsin""xcos*X(te 107

51. Integration of sin""x cos" dx: (a) by the method of undetermined .

coefficients; (6) by means of reduction formulae. .

109

62. rtan""x8ec"x(ix,jcot""xcosec*xdx 113

63. Use of multiple angles.114

54. f ^115

J a^co8^z-\- ft^sin^x

66. f ^ , f f 115Ja + bcosx Ja + bsmx

66. ie^sinnxdx, Je^cosnxdx 117

57. I sin mx cos nx (2x, (cosmxcosux(2x, I sin 7"x sin nx(Zx . .118

CHAPTER Vin

Successive Integration. Multiple Integrals

68. Successive integration 119

69. Successive integration with respect to a single independent variable 119

60. Successive integration with respect to two or more independent

variables . . . 123

61. Application of successive integration to the measurement of areas:

rectangular coordinates 126

62. Applicationof

successive integration to the measurement

ofvolumes : rectangular coordinates 128

63. Further application of successive integration to the measurement

of volumes : polar coordinates 131

CHAPTER IX

Further Geometrical Applications. Mean Values

65. Derivation of the equations of curves in polar coordinates . . 134

66. Areas of curves when polar coordinates are used : by single inte-ration

135



Xll CONTENTS

Art. Vaqm

67. Areas of curves when polar coordinates are used: by doable

integration 139

68. Areas in Cartesian co5rdinates with oblique axes.... 140

69. Integration after a change of variable. Integration when the vari-bles

are expressed in terms of another variable .

,

.141

70. Measurement of the volumes of solids by means of infinitelythin

cross-sections 142

71. Lengths of curves : rectangular coordinates 144

72. Lengths of curves : polar co5rdinates 147

73. The intrinsic equation of a curve 149

74. Areas of surfaces of solids of revolution 152

75. Areas of surfaces whose equations have the formz=f(z,y) .

156

76. Mean values 160

77. A more general definition of mean value 163

Examples on Chapter IX 164

CHAPTER X

Applications to Mechanics

78. Mass and density 167

79. Center of mass 168

80. Moment of inertia. Radius of gyration 173

CHAPTER XI

Approximate Integration. Integration bt Means of Series. Integra-ion

BY Means of the Measurement op Areas

81. Approximate integration 177

82. Integration in series 177

83. Expansion of functions by means of integration in series .

.17984. Evaluation of definite integrals by the measurement of areas J81

85. The trapezoidal rule 182

86. The parabolic, or Simpson's one-third rule 184

87. Durand's nile 187

88. The planimeter 188

CHAPTER Xn

Integral Curves

89. Introduction 190

90. Special case of differentiation under the sign of integration. . 190

91. Integral curves defined. Their analytical relations . . . 192

92. Simple geometrical relations of integral curves.... 194



CONTENTS xiii

An. Paob

03. Simple mechanical relations and applications of integral curves.

Successive moments of an area about a line....

105

04. Practical determination of an integral curve from itsfundamental

curve. The integraph 198

0" The determination of scales 200

CHAPTER Xin.

Obbiiiabt Differential Equations

06. Differential equation. Order. Degree 201

07. Constants of integration. General and particular solutions. Deri-ation

of a differential equation 202

Section L Equationsof the First Order and the First Degree

08. Equations in which the variables are easily separable . . .206

00. Equations homogeneous in x and y ..... .205

100. Exact differential equations .206

101. Equations made exact by means of integrating factors. .

.207

102. Linear equations 208

103. Equations reducible to the linear form 200

Section IL Equationsof the First Order but not of the First Degree

104. Equations that can be resolved into component equations of the

firstdegree 210

105^ Equations solvable for y 211

106. Equations solvable for x 212

107. Clairaut's equation .213108. Geometrical applications. Orthogonal trajectories . .

214

Section III. Equations of an Order Higher than the First

100. Equations of the form ^= /(") 218

dx^

110. Equations of the form ^= /(y) 218

111. Equations in which y appears in only two derivatives whose orders

differ by unity 210

112. Equations of the second order with one variable absent 220

113. Linear equations. General properties. Complementary functions.

Particular integral 222114. The linear equation with constant coefficients and second member

zero 223



XIV CONTENTS

Art. Paqi

115. Case of the auxiliary equation having equal roots . . .224

116. The homogeneous linear equation with the second member zero .225

Examples on Chapter XIII 227

APPENDIX

Note A. A method of decomposing a rational fi*action into its partial

fractions 229

Note B. To find reduction formulae for Jaf^^Cahx^y dx by integra-ion

by parts 231

Note C. To find reduction formulae for fsin""a; cos" x dx by integra-ion

by parts 233

Note D. A theorem in the infinitesimal calculus 234

Note E. Further rules for the approximate determination of areas.

235

Note F. The fundamental theory of the planimeter . . . .237

Note G. On integral curves 240

1. Applications to mechanics 240

2. Applications in engineering and in electricity . .242

3. The theory of the integraph 244

Figures of some of the curves referred to in the examples . . .240

A short table of integrals 249

Answers to the examples .263

Index 285



INTEGRAL CALCULUS

CHAPTER I

INTEGRATION A PROCESS OF SUMMATION

1. Uses of the integral calculus. Definition and sign of integration.

The integral:calpulus can be used for two purposes, namely:

(a) To find the sum of an infinitely large number of infinitesi-als

of the form f(x)dx ;

(b) To find the function whose differential or whose differential

coefficient is given ; that is,to find an anti-differentialor an anti-

derivative.

The integral calculus was invented in the course of an en-deavor

to calculate the plane area bounded by curves. The area

was supposed to be divided into an infinitely great number of

infinitesimal parts, each part being called an element of the area ;

and the sum of these parts was the area required. The process

of finding this sum was called integration, a name which implies

the combination of the small areas iutd a whole, and hence the

sum itself was called the whole or the integral

From the point of view of the firstof the purposes justindi-ated,

integration may be defined as a process of stimmation. In

many of the applications of the integral calculus, and, in particu-ar,

in the larger number of those made by engineers, this is the

definition to be taken. On the other hand, however, in many

problems it is not a sum, but merely an anti-differential,that is

required. For this purpose, integration may be defined as an

operation which is the inverseofdifferentiation.t may at once be

1



2 INTEGRAL CALCULUS [Ch. I.

stated that in the course of making a summation by means of the

integral calculus it will be necessary to find the anti-differential

of some function ; and it may also be said at this point, that the

anti-differentialcan be shown to be the result of making a sum-mation.

Each of the above definitions of integration can be de-ived

from the other. These statements will be found verified in

Arts. 4, 11, 13.

In the differentialcalculus, the letter d is used as the symbol

of differentiation,and df(x)is read "the differentialoif(x)," In

the integral calculus the symbol of integration is*

|, andif(x)x

is read "the integraloif(x)dx,'^

The signs d and I are signs of

operations ; but they also indicate the results of the operations

of differentiation and integration respectively on the functions

that are

written afterthem.

The principal aims of this book are : (1)to explain how sum-mations

of infinitesimals of the form f(x)dx may be made ; (2)to

show how the anti-differentialsof some particular functions may

be obtained.

2. Illustrations of the summation of infinitesimals. Two simple

illustrations of the summation of an infinitenumber of infinitely

small quantities will now be given. They will help to familiarize

the student with a certain geometrical principle and with the

fundamental theorem of the integral calculus, which are set forth

in Arts. 3, 4. The method employed in these particular instances

is identical with that used in the general case which follows them.

* This is merely the long Sj which was used as a sign of summation by

the earlier writers, and meant** the sum of." The sign Jwas first employed

in 1675, and is due to Gottfried Wilhelm Leibniz (1646-1716),ho invented

the differential calculus independently of Newton. The word integral ap-eared

first in a solution of James Bernoulli (1654-1705),which was pub-ished

in the Acta Ernditorum, Leipzig, in 1690. Leibniz had called the

integral

calculus calculus summatoriu$,

but in 1696 the term calculus in-

tegralis was agreed upon between Leibniz and John Bernoulli (1667-1748).

See Cajori,History of Mathematics^ pp. 221, 237.



1-2.] INTEGRATION A PROCESS OF SUMMATION

(a) Find the area between the line whose equation is y = mz,

the a?-axis,and the ordinates for which a? = a, a: = 6.

Let OL be the line y = mx', let OA be equal to a, and OB to 6,

and draw the ordinates APy BQ. It is required to find the area

of APQB, Divide the segment AB into n parts, each equal to

L

Y

Aa?; and at the points of section ^" A^y """, erect ordinates AiP^,

AiP^ ..., which meet OL in Pi, P^ .... Through P, Pi, Pg, ..., Q,

draw lines parallel to the axis of x and intersecting the nearest

ordinate on each side, as shown in Fig. 1, and produce PBi to

meet BQ in C

It will firstbe shown that the area APQB is the limit of the

sum of the areas of the rectangles Pî, Pi^2" """" when n, the

number of equal divisions of AB, approaches infinity, or, what

is the same thing, when Aa? approaches zero. The area APQB

is greater than the sum of the "inner^^

rectangles Pî, Pi^2" """ ;

and itis less than the sum of the "outer "rectangles ^Pi, AiP^, """.

The difference between the sum of the inner rectangles and the

sum of the outer rectangles is equal to the sum of the small rec-tangles

PPi, Pi Pg, ."".

The latter sum is equal to

îPiAa; + J5,P2Aa?4-- +J5"QAaj;

that is,to {B^Pi+ JâPj 4- """ + B^Q) Aa?, which is CQ Aa?.

INTEGRAL CALC. " 2



4"

INTEGRAL CALCULUS [Ch. I.

This may be briefly expressed,

S^Px-

SPî = SPP,

= CQ Aa;.

When Aa; is an infinitesimal, the second member of this equa-ion

is also an infinitesimal of the firstorder ; therefore, when Aa;

isinfinitelysmall the limit of the difference between the total areas

of the inner and of the outer rectangles is zero. The area APQB

lies between the total area of all of the inner and the total area

of all of the outer rectangles. Hence, the area APQB is the

limit both of the sum of the inner rectangles and of the sum of

the outer rectangles as Aa? approaches zero. Each elementary

rectangle has the area y Aa?, that is mx Aa;, since y = mx. The

altitudes of the successive inner rectangles, going from A towards

B, are ma, m (a-|-a;), (a-f2 Aa;),"

", m (a-h(w" 1)Aa;).Hence,

Area APQB = limit^3.ô^{a Aa; + (a+ Aa;)a; -f(a+2 Aa;)a; + " " "

+ (a+ n-lAa;)Aa;{*

=

limit^^fpn\a+(a+ Ax)'i-(af 2Aa;)-f""

+ (a+ w " 1 Aa;)Aa;.

Addition of the arithmetic series in brackets gives

Area APQB =

limit^^ô^^^2a

-f(n- 1)Aa;}

= limit^^ô^" "~^^{6-f a " Aa;},ince n Aa;= 6" a,

="(f-i)" The symbol Aa;=0 means

"

when Ax approaches zero as a limit." It is

due to the late Professor Oliver of Cornell University.



2.] INTEGRATION A PROCESS OF SUMMATION

In this example the element of area is obtained by taking a

rectangle of altitude y, that is,ma?, and width Ax, and then letting

Aoj become infinitesimal.

The expression n = oo may be used instead of Aa? = 0, since

Aaj =

h " a

It may be noted in passing, that ifthe anti-differentialof rnxdac,

namely^'^j be taken, and h and a be substituted in turn for x, the

difference between the resulting values will be the expression

obtained above.

(6)Find the area between the parabola y"^, the aj-axis,and

the ordinates for which x= a, x= b.

Let QiOQ be the parabola y = x*; let OA be equal to a,

antj

OB

to b. Draw the ordinates AP, BQ. It is required to find the

area APQB, Divide the segment AB into n parts each equal to

Fig. 2.

Aa?, and at the points of division A^ A2, """, erect ordinates AiP^,

A2P2, """. Through P, Pi, P2, """, draw lines parallel to the axis

, of X and intersecting the nearest ordinates on each side, as in

Fig. 2. It can beshown,

inthe

same

wayas in

the previousillus'

tration, that the area APQB is equal to the limit of the sum of



6 INTEGRAL CALCULUS [Ch.I.

the rectangles P-4" FiA^, """, Pn-i-B, when Aoj approaches zero.

This area will now be calculated.

The area of any elementary rectangle is yAa?, that is oÂa;,

since y = aJ* ; and the altitudes of the successive rectangles, going

from A toward B, are a*, (a+ ^xf,(a-f2 Aa?)*,"". Hence,

Area APQB = limit^ô{"*â;-h(a

4- Aaj)*Aa?-|-(a+2Aa?)*Aa?f-..

+ (a4-w " lAaj)2Aa;|

= limit^ô{a'+(a Aaj)"-fa+ 2 Aaj)*

+ (a-|-n~lAaj)2|Aa?

= limit^ôl^'+ 2 a Aaj(l+2-h3H-...+w-l)

+ (Aaj)Xl'H-2"3*+ ... +ir^*)}Aa?.

It is shown in algebra that the sum of the squares of the first

n natural numbers, 1^ 2^ 3^ ..., n\is ^(^ + l)(2tt l\ r^^^

6

application of this result to the sum of squares in the second

member of the last equation and the summation of the arith-etical

series

1, 2, 3, """

(n"

1),gives

Area APQB = limit^ô^ ^x"a^-\- an^x " a Ax

+

(Aa,)'("-l)(But nAa? = 6 " a;

and hence, a* + an Aa;-f-

" (Aa;)*^(a*-f-a6 -f 6*).

Hence,ÎPQB

= limit^ô("-a)i""*+^+"'

" ^^

3 3



2-3.] INTEGRATION A PROCESS OF SUMMATION

In this example, the element of area is obtained by taking a

rectangle whose area is aÂic and then letting Ax be an infinitesi-al.

Here also, it may be noted in passing, that if the anti-

differential of a^dx, namely "

,be taken, and b and a substituted

o

in turn for ", the difference between the resulting values will be

the expression justderived for the area.

3. Geometrical principle. Let f(x) be a continuous function

of X, and let FQ be an arc of the curve whose equation is

Fio. 8.

y=f(x).Draw the ordinates AP, BQ, and suppose OA = a,

OB = b. It is required to find the area APQB ; that is, the

area between the curve, the oj-axis,and the ordinates AP, BQ,

Divide the segment AB into n parts, each equal to Ax, and at

the points of section ^j, ^2?"""" erect the ordinates A^Pi,

-^jPj,"""

to meet the curve in P^ Pg, "... Through P, Pi, Pg, """, draw

lines parallel to the cr-axis and intersecting the nearest ordinate

on each side, as in Fig. 3, and produce PBi to meet BQ in (7.

It will nT)w be shown that the area APQB is the limit of the

sum of the areas of the rectangles PAi, P1A2, """, when the num-




ber of equal divisions of AB is made infinitely great, that is,when

Ax is made infinitely small. The area APQB is greater than the

sum of the areas of the inner rectangles PAi, PiA^, """, and less

than the sum of the areas of the outer rectangles APi, AiP^, """.

That is,

^PA,"APQB"^P,A,

The difference between the sum of the outer rectangles and the

sum of the inner rectangles is equal to the sum of the small

rectangles PPi, P1P3, """ ; that is,

= BiPi^x + J^jPjAa? 4- " + B^Q Aoj

= (APi + ^2^2 + - + B,Q)Ax

=zCQAx.

The difference, CQ Ax, can be made as small as one pleases by

decreasing Ax, Therefore, since the area of APQB always lies

between the areas of the inner and outer series of rectangles,

and since the difference between these areas approaches zero as

its limit, the area APQB is the limit both of the sum of the

inner rectangles, and of the sum of the outer rectangles.

Therefore, the area induded between the curve whose equation is

y =f(x),the Qc-axis,and a pair of ordinates, is the limit

of the sum

of the areas of the rectangles whose bases are successive segments of

the part of the x-axis intercepted by the pair of ordinates, and whose

altitudes are the ordinates erected at the points ofdivision

of the

X-axis, as the bases approach zero,

4. Fundamental theorem. Definite integral. Since the equa-ion

of the curve, an arc of which is given in Fig. 3, isy=f(x),

the heights of the successive inner rectangles, going towards the

right from A, are

/(a),fia + Ax),f(a + 2Ax),"',f{a+(n-l)Ax).



3-4.] INTEGRATION A PROCESS OF SUMMATION 9

Hence,

Area APQB = limitAojô{/(a)a?

-f

/(a + Aa?)a:

+/(a + 2 Aaj)oj + """ +f(a + w - 1 Aoj)Aoj}. (1)

The second member of (1)is the limit of the sum of the

values, infinite in number, that f(x)^x takes as x varies by

equal increments Aoj from x = a to x = b, when Aa; is made

infinitesimal. This limit may be indicated by

Limits

iO/^/(")A"5*

In the integral calculus this is more briefly indicated by

prefixing to f(x)dx the sign j, at the bottom and top of which

are respectively written the values ôf x at which the summa-tion

begins and ends ; thus :

x = h

Cf(x)x.

x=a

An abbreviation for this form is

i'^f(ix)dx.

(2)

This is read, "the integral of f(x)dx between the limits

a and 6." The initial and final values of x, namely, a and 6,

are called the lower and upper limits respectively of the inte-ral.*

The differential f(x) dx is called an element of the

integral. It evidently represents the area of any one of the

component infinitesimal rectangles of altitude f{x) and infini-esimal

base dx. In the same way that dx is a differential of

* This manner of indicating the limits between which the summation

is to bemade

bywriting

the lower limitat

the bottom

andthe

upperlimit at the top of the integration sign, is due to Joseph Fourier

(1768-1830).




the distance along the aj-axis,so f{x)dx, or y dx, as it is usually

written, is a differential of the area between the curve and

the axis of x.

The limit of the sum in the second member of (1),which is

indicated by the symbol (2),will now be obtained. Suppose

that f(x)dx is d"f"(x),hat is, suppose that 4"{x)is the antir

differentialf f(x)dx. Then, by the fundamental principle of

the differential calculus,

in which e is a quantity that varies with x and approaches

zero when Aa; approaches zero. On clearing of fractions and

transposing, the latter equation becomes

/(a?)a?= "^(a? Aa;) "l"(x)e Aa?. (3)

On substituting in (3) the values of x at the successive

points of division between A and B at intervals equal to Aa?,

the following equations are obtained:

/(a)Aa? =

" (a+ Aa?) ^ (a) Cq Aa?,

/(a -f Aa?)a; =

" (a-f2 Aa?) " (a-f A")

" î Aa?,

/(a -f2 Aa?)a;= "^(a-f

3 Aa?) "^(a-f2 Aa?) ejAa?,

f(b " Aa?)

a? =

" (6) " (6 Aa?) e^_i Aa?,

inwhich each of the e's approaches zero when Aa? approaches

zero. The sum of the first members of these equations is

equal to the sum of the second members; that is,

y^f(x)^x"l"(b)"^(a) (eo ci + - + c"_i)â?-

x = a

Of the quantities Cq, ei, """e"_i, suppose that e^ has an absolute



4.] INTEGRATION A PROCESS OF SUMMATION 11

value E not less than that of any one of the others. It E ^

substituted for each of the e's, the term

becomes nE^x, or (6"

a)Ej

since n^x^h " a. Hence, ''

^/(aj)Aa:="^(6)-"^(a).

a quantity which is not greater

than (6"

a)E, and which ap-roaches

zero when E ap-roaches

zero, that is, when

Ax approaches zero.

Therefore, on letting Aa? approach zero, there will be obtained,

X /(aj)"laj +(6)- + (a).

Hence, the mm or irUegrcU, I f(x)dx, which is the 8um of aU the

vcUueSy infiniten number, thatf(x)dx takes as x varies by infinites-

imal increments from a to b, is found by obtaining the anti-differntial

"l"(x)off(x)dx, and subtraîng the value of ^ (x)for x = a

from its value for a? = 6. The following notation is used to indi-ate

these operations :

J^/(x)(te[^(a;)J'=^(6)-4)

"It will be shown in Art. 9, that if dp(x) =f(x)dx, the anti-differential

of /(x) (te is 0 (") + c, in which c is an arbitrary constant. Hence, equation

(4)should be written

y^/(x)(to[0(a;)+c]*.

Since the same c is used when a and b are substituted for x, this becomes

JyCx)"te0(6) -0(a).as above.




The sum | f(x)dx is called a definitentegral because it has a

definite value, the limit of the series in the second member of

(1). Since the evaluation of this definite integral is equivalent

to the measurement of the area between the curve y = /(a?),he

ic-axis,and the ordinates at a; = a, a; = 6, the area may be regarded

as representing the integral. It follows from the result (4)that

a definite integral may be regarded as either :

(1)The limit of the sum of an infinitely large number of infini-esimal

quantities of the form f{x)dx taken between certain

limits; or,

(2)The difference of the values of the anti-differential of

f(xydx at each of these limits.

If f(x)is any continuous function of a?,f(x)dx has an anti-differ-ntial.*

However, the deduction of the anti-differential is often

impossible, and in any case, is less simple and easy than the

process of differentiation.f

Many of the practical applications of the integral calculus,

such as finding areas, lengths of curves, volumes and surfaces of

solids, centers of gravity, moments of inertia, mass, weight, etc.,

consist in making summations of infinitely small quantities. The

integral calculus adds these infinitesimal quantities together

and gives the result. It has been observed that in order to

obtain the sum of infinitesimal areas, etc., the anti-differential

of some function is required. Accordingly, a considerable part of

any book on the integral calculus is devoted to the exposition of

methods for obtaining the anti-differentials of functions which

frequently appear in the process of solving practical problems.

* The truth pf this statement, for all the ordinary functions, will appear

in the sequel. A proof applicable to all forms of continuous functions is

given in Picard, Traite d*Analyse^t. I., No. 4.

t The phrase "to find the anti-differential" means to deduce a. flnite

expression for the anti-differentialin terms of the well-known mathematical

functions. In cases in

whichthe

anti-differentialcannot be thus

obtained,an

approximate value of the definite integral can be found by the methods dis-ussed

in Arts. 84-88. A short inspection of these articlesmay be made now.



4-5] INTEGRATION A PROCESS OF SUMMATION 18

5. Supplement to Art. 3. In proving the principle of Art. 3,

the arc PQ in Fig. 3 was used. If the arc of the given curve

has the form and position in Fig. 4, the proof of the principle

is as set forth in Art. 3. If the arc has the form and position in

T

Fig. 6.

Fig. 5, and thus has maximum and minimum values of the

ordinate, the principle stillholds. This can be seen by drawing

the maximum and minimum ordinates that come between AF

and BQ, and remarking that the principle holds for the several

successive parts APPiAi, AiPiPiAy """.

Suppose that the curve has the form and position in Fig. 6.

The area of the part APAi is the limit of the sum of the ele-entary

areas f(x)^x when Ax approaches zero and x varies

from OA to OAi ; or, in other words, theârea

of APAi is the

limit of the sum of the elementary areas f(x)dx as x varies from

OA to OAi. Similarly, the area of A1TA2 is the limit of the sum

of the elementary areas f(x)dx ss x varies from OAi to OA^f and

the area A^QB is the limit of the sum of the elementary areas

f(x)dx ss X varies from OA2 to OB. In APAi and A2QB, the

ordinates that represent the values of /(a?)re positive, while in

A1TA2 the ordinates are negative. Since x is taken as varying

from left to right, dx is always positive. Accordingly, areas such

as APAi, AiQB, which lie above the aj-axis,have a positive sign,

and areas such as AiTA^ which lie below the avaxis, have a neg-tive

sign. This example shows that in the case of a curve that

crosses the avaxis, the method of summation by means of the

integral calculus gives the algebraic sum of the areas that lie




between the curve and the aj-axis, the areas above the a?-axis

being given a positive sign, and those below receiving a nega-ive

sign.

If the total absolute area between the curve and the axis of

X is required, the portions APjX\, A^TA^, A^QB, should be found

separately.

Note. If n is a constant not equal to " 1, the anti-differential of u*du is

^; for, differentiation of the latter gives u^du,

n

+1

Ex. 1. Find the arfea between the curve whose equation is y = x', the

X-axis and the ordinates for which x = 1, x = 4.

By Art 4, the area required = Tx^dx

= 63| units of area.

Ex. 2. Find the area between the parabola 2y = 6x^, the x-axis and the

ordinates for which x = 2, x = 6. Ana, 97^ square units.

Ex. 8. Find the area between the line y = 4 x, the x-axis and the ordi-ates

for

whichx = 2, x = 11. An8, 2*34

square units.

Ex. 4. Find the area between the parabola 2 y = 3 x^, the x-axis and the

ordinates for which x = " 3, x = 5. Ans. 76 square units.

Ex. 5. Find the area between the line y = 5 x, the x-axis and the ordinate

for which x = 2. Ans. 10 square units.

Ex. 6. Find the area between the line y = 6x, thex-axiis and the ordinates

for which x = " 2, x = 2. Ans, 0.

6. Geometrical representation of an integral. It is necessary to

perceive clearly that a definite integral, whether it be the sum

of an infinite number of infinitesimal elements of area, length,

volume, surface, mass, force, work, etc., can be represented graphi-ally

by an area. For instance, in order to represent the definite

integral



5-7.] INTEGRATION A PROCESS OF SUMMATION 15

choose a pair of rectangular axes, plot the curve whose equation

and draw the ordinates for x = a, x=: h. It has been shown in

Art. 4 that the area between this curve, the a?-axis,and these

ordinates has the value of the definite integral above. Hence,

this area can represent the integral. This does not mean that

the area is equal to the integral, for the integral may be a length,

a volume, etc. The area can be taken to represent the integral,

because the number that indicates the area is equal to the number

that indicates the value of the integral. That an integral may

be represented geometrically by an area is at the foundation

of some important theorems and applications of the integral

calculus.

7. Properties of definite integrals. In Art. 4 it was shown that if

the definite integral, jf(x)dx= if b) ifâ).

From this, the firstof the following properties is immediately

deducible. The second and third properties depend upon Art. 6.

(a)"f(x)dx^--jy(x)dx.

This relation holds since the second member is " } (a) " (6)|;

that is,^(6)" 4"{a),Hence, the algebraic sign of a definite inte-ral

is changed by an interchange of the limits of integration.

%/a *ya %/c

Let the curve whose equation isy=f{x)

be drawn; and



16 INTEGRAL CALCULUS [Ch.I.

let ordinates AP, BQ, CR be erected at the points for which

x= a, x=by 0? = c. Since

area APQB = area APBC

+ area CBQB,

rf(x)dx cy(x)dx+Cf{x)dx,%/a "/a "/"

It does not matter whether c is

-Xbetween a and h or not. For, sup-ose

OC = c', and draw the ordi-ate

OW ; then

C B C

Fig. 7.

area APCIB = area APE'C - area BQR'C ;

f(x)dx= I f(x)dx" I f(x)dx,

which, by (a),

=J^(aj)cte

+j[(a?)a?.

Therefore a given definite integral may be broken up into any

number of similar definite integrals that differ only in the limits

between which integration is to be performed.

(c)Construct APQB as in (6)to represent the definite integral

rf(x)dx.Then

f f(x)dx = area APQB

Q

r= area

of

a

rectangle whoseheight CB is greater

than AP and less than

BQ, and whose base

isABy

= AB'CB

=

(6-a)/(c),

OC being equal to c.



7.] INTEGRATION A PROCESS OF SUMMATION 17

Therefore /(c)= ^ "-

0"

a

The function /(c)is called the mean value of f{x)for values of

X that vary continuously from a to 6. This mean value may be

defined to be equal to the height of a rectangle which has a base

equal to 6 " a and an area that is equivalent to the value of the

integral. Thesubjectof mean values is discussed further in

Arts. 76, 77.



CHAPTER II

INTEGRATION THE INVERSE OF DIFFERENTIATION

8. Integration the inyerse of differentiation. In Art. 1, two

definitions of integration were indicated, namely :

(a)Integration is a process of summation ;

(b)Integration is an operation which is the inverse of differen-iation.

The first definition was discussed in Chapter I. In this and

the next following article,integration will be considered from the

point of view of the second definition.

The differential calculus is in part concerned with finding the

differential or the derivative of a given function. On the other

hand, the integral calculus is in part concerned with finding the

function when its differential or its derivative is given. If a

function be given, the differential calculus affords a means of

deducing the rate of increase of the function per unit increase of

the independent variable. If this rate of increase of a function

be known, the integral calculus affords a means of finding the

function.

Ex. 1. A curve whose equation is y = 4 ac^ isgiven ; and the rate of increase

of the ordinate per unit increase of the abscissa is required.

Since y = 4 x^,

^= 8x.

dx

This means that the ordinate at a point whose abscissa is x is increasing 8 z

times as fast as the abscissa. If this rate of increase remains uniform, the

ordinate will receive an increase of 8 x when the abscissa is increased by

unity. This determines the direction of the curve at the point.

18



8.] INTEGRATION INVERSE OF DIFFERENTIATION 19

On the other hand, suppose that at any point on a curve, it is known that

dx

and let the equation of the curve be required. It evidently follows from this

equation that

in which c is an arbitrary constant. The constant c can receive any one of

an infinite number of values ; and hence, the number of curves that satisfy

the given condition is infinite. If an additional condition be imposed, for

example, that the curve pass through the point (2,3), then c will have a

definite corresponding value. For, since the point (2,3) is on the required

curve,

3 = 4.22 + c,

and accordingly, c = " 13.

Hence, the equation of the curve that satisfiesboth of the conditions above

given is

y = 4xa-13.

Ex. 2. In the case of a body falling from rest under the action of gravity,

the distance s through which itfallsin t seconds is a constant, approximately

16 times ^ feet ; find the velocity at any instant.

Here, " = 16f2,

andhence, ^ = S2t;

that is,the velocity* in feet per second at the end of t seconds is 32 1.

On the other hand, suppose it is known that in the case of a body falling

from rest, the velocity in feet per second is 32 multiplied by the time in

seconds since motion began. Let the corresponding relation between the

distance and the time be required.

daHere, it is known that " = 32 ".

'

dt

" If a body moves in a straight line through a distance As in a time M, and

if its average velocity be denoted by v,

As

At

As At approaches zero. As also approaches zero, and the velocity approaches

dathe definite limiting value ^"

dt

INTEGRAL CALC. " 3



20 INTEGRAL CALCULUS [Ch. II.

It is obvious that the solution of this simple differential equation is

(1) "=:16"2 + C,

inwhich

c is an

arbitrary constant.Tliis

resultis indefinite. By

the condi-ionsof the question, however, " = 0 when i = 0.

Hence, substituting in(l), 0 = 0 + c;

whence, c = 0,

and " = 16"2

is the definite solution.

The distance fhroughwhich

the body falls can

alsobe deter-ined

by the method of summation employed in the firstchapter.

Let the number 32 be denoted by g. The distance passed over in

any time is equal to the product of the average velocity duiing

that time and the time. The time t may be divided into n equal

intervals A^, so that ^ = nAf. The velocity at the beginning of

therth

interval is

(r 1)Af,

and atthe

end ofthe interval is

rg A^. Hence, the distance passed over in the interval liesbetween

(r-l)g{My^XLdLrg{M)\

On finding similar limits for the distance passed over in the

case of each of the intervals and adding, it will be found that the

total distance passed over lies between

[0+ 1 + 2 + - + (n - V)-\g{Myand [1+ 2 + - + n^gi/Hty-,

that is, summing these arithmetical series, the distance passed

over lies between

Since A^ =-, the distance lies betweenn

2 2n 2^2n'

and the distance is the common limit of these two expressions,

when A^ approaches zero, that is, when n approaches infinity.

Hence, 8 = ^gt^,




Sometimes the anti-differential(orthe anti-derivative)f a func-ion

is wanted for its own sake alone, as in the illustrations

given above ; and sometimes it is desired for further ends, as, for

example, in the process of making a summation by means of

the integral calculus in Art. 4. The anti-differentialis called the

integraly the process of finding it is called integration, and the

symbol of integration is the sign I.

Thus, if the differential of

" (x)is/(")dx, which is expressed by

d"^(aj)=/(aj)daj, (1)

then Cf(x)dx"fi(x),

" (2)

Equation (1)may be read" the differential of " (x)

isf(x)dx ;"

equation (2)may be read" the function of which the differential

is f{x)dx, is"fi")."

For brevity, the latter may be read"the

integral of f(x)dxis "^(aj).""

Memory of the fundamental formulae of differentiation will

carry one far in the integral calculus. For instance, since rfa^

is 4a^(22r, I4tx^dx is ic*; since dsina; is cos a; da:, |cos a:da? is

sin X, The beginner will see the necessity of having ready com-mand

of the formulae for differentiation, since they will be

employed in the inverse process of integration.t Differentiation

* The origin of the terms integral, integration, and of the sign i has

been given in Art. 1. Instead of the sign 1,the symbols d~^ and 2"-i are

sometimes employed:thus,

(?-y(x)(to,hich is read **the

anti-differentialof f(x)dx,

' '

and D-^f{x) which is read,* *the anti-derivative of f(x)/

' In the

case of the second definition of integration, the use of the symbols d~\ D~^,

is more logical than the use of \. The latter sign is, however, fii-mly estab-ished

in this connection. It may be remarked that the differential is more

frequently written than is the derivative of a fimction.

t The expressions Xx^dx, d''\x^dx),D'îxP) are equivalent. The in-erse

process of

integration is notalways practically possible (see

Art.

4).Art. 81 may be referred to for examples of differentials whose integrals can-ot

be expressed in a finiteform.



22 INTEGRAL CALCULUS [Ch. IJ.

of both members of (2)gives*

whence, by (1),=/(^)

^^"

Therefore, d neutralizes the effect of 1.

It will be shown in

the next articlethat jd"^(aj)ay have values different from" (").

9. Indefinite integral. Constant of integration. Since d(Qi^-{-c)

is 4aj'da?, c being any constant, l4:a^dxis a^ + c. The integral

given in Art. 8 comes from this on making c zero. But c may

be given any other value that does not involve x. Hence, i 4ta^dx

is indefinite so far as an arbitrary added constant is concerned.

In general,

if d"tt(x)=f(x)dx, (1)

then ff(x)dx=if"{x)c, (2)

in which c is any constant ; for differentiation of the members of

(2)shows that f{x)dx=idit"(x).ence, the integral of a given

differentials indefiniteo far as an arbitrary added constant is

concerned. Illustrations have been seen in the preceding articles.

It should be noted that the indefiniteness does not extend to

terms that contain x. In other words, a given differential can

have an infinitenumber of integrals that correspond to the infi-ite

number of valuesthat an

arbitrary constantcan take, but

any two of these integrals differ only by a constant. For

instance,

/'x-^l)dx= ^-\-x-\-Ci.

But on substituting zfovx-h 1, and consequently, dz for dx,

J(x-^l)dx=fzdz^^-]^c,==^^^^-hc,^



8-10.] INTEGRATION INVERSE OF DIFFERENTIATION 23

These two integrals agree in the terms that contain x. When

an integration is performed, the arbitrary constant should be

indicated in the result ; or, if not indicated, it should be under-tood

to be there. The second member of (2)is usually called

the indefinitentegral of f(x)dx, and c is said to be the constant

of integration. When the constant of integration has an arbitrary

value, that is,when no definite value has- been assigned to it,the

integral is called also a general integral ; on the other hand, when

the constant of integration is given a particular value, the integral

is said to be a particular integral. For instance, the general

integral (and indefinite integral)f oj*da? is â^ + c in which

c is arbitrary. A particular integral of a^dx is obtained by

giving c any one of an infinity of possible values, say 6, " 5, ^.

Thus \3ii^6y \iii^5, îx^-hi are particular integrals. In

practice the value of the constant may be determined by the

special conditions of the problem.

10. Geometrical meaning of the arbitrary constant of integration.

If |=-^(*)' (1)

then y = I^\^) ^y

that is, y = F(x)+ c, (2^

in which c is an arbitrary constant of integration. Suppose that

c is

given particular values, say

8, " 3, etc. ;and

let the curves

whose equations are

y = i^(aj)+8,|

etc., etc.,

be drawn. All of these curves have the same value of -X ; thatdx

is, the same direction, for the same value of a?. Also, for any



24 INTEGRAL CALCULUS [Ch.il

Fig. 9.

two of the curves the difference in the lengths of their ordinates

remains the same for all

values of x. For example,

for any value of x, the dif-erence

between the lengths

of the ordinates of the two

curves whose equations are

given in (3)is 8-(-3),

or 11. Hence, all the

curves, whose equations are

of the form (2),hus differ-ng

merely in the c's, can be obtained by moving any one of the

curves vertically up or down. The particular value assigned

to c merely determines the position of the curve with respect to

the oj-axis,andhas

nothingto do

withits form. Fig. 9 illustrates

this.

11. Relation between the indefinite and the definite integral. In

Art. 4 it has been seen that if d"l"x)=f(x)dx, the sum or inte-ral

of f{x)dx for all values of x from x = a to x = b, satisfies

the relation

"f(x)dx="f^(b)^"k(a)1)

If the upper limit be variable and be denoted by x,

"f(x)d^"f^(x)^4"(a). (2)

If the lower limit a be arbitrary, "

" (a)may be represented

by an arbitraiy constant c, and (2)becomes

But

Jf{x)dx"l"(x)+c.

Jf(x)dx=îf"(x)c

(3)

(4)



10-12.] INTEOBATION INVERSE OF DIFFERENTIATION 26

Hence, an indefinite integral is an integral whose upper limit

is the variable and whose lower limit is arbitrary. The first

member of (4)may be considered an abbreviation for the first

member of (3). The indefinite integral may. therefore be re-garded

as obtained by a process of summation.

12. Examples that involve anti-differentials. This article is in-erted

for the purpose of giving simple, typical examples of a

practicalkind, in

which anti-differentials are requiredfor

pur-oses

other than that of summation. These illustrations will

also involve the determination of constants. In many applica-ions

of the calculus, two kindsof constants must be distinguished,

namely, those which are constants of integration, and those

which are given distances, angles, forces, etc. ; for example, the

constantg,

in Ex. 2, Art. 8,

and

h, 6, k, a, in Ex. 2 below. Rec-angular

coordinates are used in the following exercises.

Ex. 1. Determine the equation of the curve at every point of which the

tangent has the slope J. Determine the equation of the curve which passes

through the point (2, 3) and also satisfies the former condition. The

slope of a curve y "f(x) at any point (x,y)is

-^.*Hence, by the given

condition,

(1)^

=1

^^dx 2

Adopting the differentialform, dy = idXy

and integrating, (2) y = ^ x + c,

the equation of a straight line. Now c, the arbitrary constant of integration,

which in this case represents the intercept of the line on the j/-axis,can take

an infinite number of values. The first condition is therefore satisfied by

each and all of the parallel lines of slope ^.

If, in addition, the line is required to pass through the point (2,3),then

X = 2, y = 3, satisfy (2), and S = \ -2 -\-c. From this, c = 2. Hence, the

curve that satisfiesboth of the given conditions is the line whose equation is

y = ix-^2.

* By the slope of a curve at any point is meant the tangent of the angle

that the tangent line to the curve at the point makes with the x-axis.




Ex. 2. Determine the equation of the curve that shall have a constant

subnormal. Also, determine the curve which has a constant subnormal and

passes through the two given points (o,h), (",k); and find the length of its

constant subnormal.

Let A, B, be the given points (o,h),(6,k),and let P be any point (x,y)on the curve. Suppose that PT is the tangent at P, and PH the normal.

Put the constant subnormal 3f2V equal to a.

Since the angle a = angle 0 (seeFig. 10),their sides being respectively

perpendicular,

tan a = tan $ ;

that is, " !=-:"Putting this in the differentialform,

(2) ydyâdz.

and integrating both sides,^-^c'

= ax-\- c",

whence. (3) |- ax + c,

in which c', c", are the constants of integration, and c denotes c" " c'.

Equation (3)isthe equation of a parabola, and it includes an infinitenumber

of parabolas, one for each of the infinite number of values that the arbitrary

constant c can have.

The particular curve which passes through the points (o,h) (6,k), and

has a constant subnormal is also required. Since the coordinates of these

points must satisfy (3),it follows that

andF

= ab -\-c.



12-13.] INTEGBATION INVERSE OF DIFFERENTIATION 27

These equations suffice to determine c and the length a of the constant

subnormal. On solving them, it is found that

c=iA2, a = -

26

Hence, the equation of the secondcurve required is

In the differential calculus it is shown that the length of the subnormal

is y

-^.The firstcondition might have been expressed immediately by the

equation

which is equivalent to (1)and (2).

Ex. 3. Find the curve whose subtangent is constant and equal to a. De-ermine

the curve so that it shall pass through the point (0,1).

Ex. 4. Find the curve for which the length of the subnormal is propor-ional

to (sayk times)the length of the abscissa.

13. Another derivation of the integration formula for an area.

Ill Arts. 3, 4, it was shown that the area included between the

curve y =/(a;),he avaxis, and the ordinates for x = a, x=h is

the limit of the sum of the infinitely large number of infinites-mal

quantities f{x)dx,which are successively obtained as x

varies continuously from a to b, and this limit was represented

by the definite integral I f(x)dx. The area can also be derived

by means of the second defi-ition

of integration.

Let CPQ be an arc of the

curve whose equation is y =

f(x),and let OA = a,OB = b.

Draw the ordinates AP, BQ,

Take any point S on the

avaxis at a distance x from

0, and draw the ordinate SL whose length is /(a?).Let z

denote the area of OCLS. If x or OS is increased by ST,

which is equal to Ax, and the ordinate TM be drawn, the area

z will be increased by the area SLMT. This increment will

Fro. 11.




be denoted by A2;. DrawXi? parallel to the ovaxis, and complete

the rectangle LRMV in which LB = "^x and EM= Ay.

The increment of the area,

i^z^SLMT

= rectangle SLRT-h

area LRM

= SLRT-\- an area less than the rectangle LRMV

=/(")Aa:

+ arealess

than Ay" Aa.

Hence,

A2"

=/(ic)-h something less than Ay.Ao;

When Aa? approaches zero, Ay also approaches zero ; and hence,

in the limit,

|=/(^); (1)

that is, J = y. (2)

Equation

(2)means

that the numerical value ofthe differential

coefficient with respect to the abscissa, of the area between a

curve, the axes of coordinates, and an ordinate, is the same as the

numerical value of this ordinate of the curve.

Equations (1)and (2)written in the differential form give the

differential of this area, namely,

dz=f{x)

dx, and dz = ydx, (3)

Finding the anti-differentialsin (3)gives as the area OGLS,

I dx

=

*(a?)+c,(4)

in which "^(a:)s the anti-derivative of f(x)and c is an arbitrary

:=J/(a!),




constant. If the area is measured from the j^-axis,the area is

zero when x is zero. Hence, substituting these values in (4)

0 =

"^(0)-hc,

whence, c = "

"^(0),

and (4)becomes z =

"f"(x)"^(0).

Hence, area of OCPA =

"^(a) "^(0).

Similarly, area of OCQB =

"^(6) "^(0).

Since APQB = OCQB - OCPA,

it follows that area APQB =

"^(6) "t"(a).

The expression in the second member is the same that was

found for the area in Art. 4 by means of the first definition of

integration.

If the area is measured, not from the y-axis, but from another

fixed vertical line, say the ordinate at x = m, the derivation of

equations (1)and (2)is the same as given above. In this case,

the area is zero when x = m, and hence,

0 =

"^(m)c. From this, c = "

"f"(m).

The value of c in (4)thus depends solely upon the fixed ordi-ate

from which the area is measured.

14. A new

meaning of yin the curve

whose equationis

i/=

f(x).Derived curves. It has been seen in the differential calculus

that in the case of a curve whose equation isy=f(x), at any

point on the curve the slope of the curve is-^,

the differentialdx

coefficient of its ordinate with respect to its abscissa. Art. 13,

with equations (2)and (4),hows that at any point of a curve

the length of the ordinate y is the differential coefficient with

respect to the abscissa, of the area bounded by the curve, the




axis of X, a fixed ordinate, and the ordinate at the point.* There-ore,

" if we wish to make a graphic picture of any function and

its derivative, we can represent the function either by the ordi-ate

y of a curve or by its area, while its derivative will then be

represented by its slope or ordinate respectively. If we are most

interested in the function,e usually employ the former method

(inwhich the ordinate represents the function)if in its denva-

tive, the latter (inwhich the ordinate represents the derivative).

That is,we usually like to use the ordinate to represent the main

variable under consideration.''f

For instance, suppose it is necessary to represent the function

f(x). Let the curve be drawn whose equation is

y=f(x). (1)

At any point (a?,y)

on the curve, the ordinate y represents the

value of the function for the corresponding value of x ; and the

slope -2 represents the rate of change of the function compareddx

with the rate of change of the variable x. Now let the curve be

drawii whose equation is

y=f'(^), (2)

df(x)in which /'(a?)enotes '^^

^- At any point (a?,y) on this curve.

* The remaining part of this article is not necessary for the articles that

follow. However, it may be useful for the beginner to read it,because it

may help to strengthen his grasp on the fundamental principles of the

integral calculus.

t Irving Fisher, A briefintroduction to the Infinitesimalalculus designed

especially to aid in reading mathematical economics and statistics. Art. 89.

Some readers may be interested in an application of the principle quoted

above. Professor Fisher continues: "Jevons, in his Theoryof

Political

Economy, used the abscissa x to represent commodity, and the area z to

represent its total utility, so that its ordinate y represented*

marginal

utility' (i.e.he differential quotient of total utility with reference to com-modity).Anspitz

andLieben, on the

other

hand, in their Untersuchungen

Uber die Theorie des Preises, represent total utility by the ordinate and

marginal utility by the slope of their curve."




the numerical measure of the ordinate is the same as that of the

slope of the first curve at the point having the same abscissa.

Hence, an ordinate at any point of the second curve represents

the rate of change of the function f(x)compared with the rate of

change of the variable x at this point. Also, the area between

the second curve, the axes, and the ordinate at (x,y)is

that is /(a')-/(0).

Hence, the area of the curve y=zf'(x)bounded as described

above plm a constant quantity /(O) can represent the function

For example, suppose that the function ispa^ + 4. That is,

f(x)=p:x?-\-^,

and f(x) = 2px,

The parabola y = ps? -\-4, and the line y = 2px are shown in

Fig. 12. At any point M on the avaxis, the ordinates MPj MQ

are drawn to these curves. The ordinate MP represents the

function for x= 0M\ and the slope at P represents the rate of

change of the function when x = OM, The ordinate MQ is

equal (numerically)o the slope at P\ and hence, it also repre-




sents the rate of change of the function when x = OM. More-ver,

since

area OQM = | 2px dx = pa^,

the function f(x)for x=OM is represented by the area 0Q3f +4.

Had the function been px^ (shownby the dottedcurve),he area

OQM would exactly represent the function.

To recapitulate : In the case of a function /("),if the curve

y =/(?"), (1)

and its firstderived curve y = f'(x)y (2)

be drawn, the rate of change of the function for any value of x

is represented equally well by the slope of the firstcurve and by

Y

Fi6. 18.

the ordinate of the derived curve for that value of a?; and the

function itself for any value of x is represented equally well by

the corresponding ordinate of the primary curve and by the area

of the derived curve increased by the constant quantity /(O).



14-15] INTEGRATION INVERSE OF DIFFERENTIATION 33

The derived curve (2)is called the "curveof slopes" of the

first curve. Two of these curves are shown in Fig. 13. The

horizontal scale is the same for both curves ; but the ordinates

on the original curve represent lengths while the ordinates on

the derived represent tangents of angles. At a point at which

the original curve has a maximum or a minimum ordinate, the

slope is zero; and hence, the corresponding ordinate on the

derived curve is zero. Conversely, when the derived curve

crosses the avaxis, the corresponding ordinate of the original

curve is a maximum or a minimum.

ISi Integral curves. Let the curve whose equation is

y=f(^) (1)

be drawn. Suppose that the anti-derivative of f(x)is "l"{x)and

draw the curve whose equation is

y=fjf(x)dx,(2)

that is, y =

"f"(x)"^(0),

or, briefly, y = F(x), (3)

The curve whose equation is (2)or (3)is called the firstinter

grcU curve of the curve (1). It is evident that

^=^=/('). "

The following important properties can be deduced from equHr

tions (1),(2),(4).

(a)For the same abscissa x, the number that indicates the

length of the ordinate of the first integral curve is the same as

the number that indicates the area between the original curve,

the axes, and ordinate for this abscissa. Therefore, the ordinates

of the firstintegral curve can represent the areas of the original

Qurve bounded as above described.



u INTEGRAL CALCULUS [Ch. II.

(6)For the same abscissa a?, the number that indicates the

slope of the first integral curve is the same as the number that

indicates the length of the ordinate of the original curve. There-ore

the ordinates of the original curve can represent the slopes

of the firstintegral curve.

Example. The line whose equation is

y=px

has for its firstintegral curve the parabola whose equation is

.y =

jpxdx,7?

that is, yz=p

At any point M on the ic-axis,OM being equal to x^^ say, erect

the ordinates MP, MPu to the line and the parabola. The same

/y.2

number, namely p-^,indicates both the length of the ordinate

MPi and the area OPM\ and the same number, namely pxi, indi-ates

both the length of the ordinate MP and the slope of the

tangent at Pi. This is true for the pair of ordinates erected at

every point on the a:-axis.

In like manner the curve whose equation is (2)has a first

integral curve. The latter is called the second integral curve

for the curve of equation (1). This second integral curve has a




first integral curve which is called the third integral curve of

(1),and so on. There is thus a series of successive integral

curves for

any givencurve. For instance, the

second

integral

curve of the line y=px is the curve whose equation is

y

that is, y = P-^-

This curve is shown in the figure. Thesubjectof successive

integral curves has very important applications in problems in

mechanics and engineering. Accordingly, an exposition of their

properties and uses is given in Chapter XII.

16. Summary. This and the preceding chapter have been con-cerned

with showing by statement and examples that integration

may be regarded in two ways :

(1)As a process of summation, in which I f(x)dx denotes the

limit

ofthe sum indicated by

^ f{x)^x,whenAa:

approacheszero; "="

(2)As an operation which is the inverse of differentiation, in

which If(x)dx denotes d'^lf^x)x']r D-^f(x); that is,denotes

the anti-differentialoif(x)dx, or, what is the same thing, the anti-

derivative of f(x).

It may be remarked that the rules of integration are all derived

from the latter point of view. Both of these conceptions of

integration are employed in problems in geometry, mechanics, and

other subjects.The first view of integration is necessary to a

clear understanding of the application of the integral calculus to

the solution of certain problems; and, on the other hand, the

second view is necessary to a clear understanding of the use of

the calculus in the solution of certain other problems.

INTEGRAL CALC. " 4



CHAPTER III

FUNDAMENTAL RULES AND METHODS OF

INTEGRATION

17. In Chapters I. and IT., the two purposes of integration

were set forth; and definitions of integration based upon these

purposes were given with illustrative examples. Relations be-ween

the definitions were also pointed out, particularly in Arts.

11, 13. It was also shown that in the process of making an

integration, whatever the

objectmay be, it is necessary to find

an anti-differential or an anti-derivative of some function. A

general method of differentiation is given in the differential

calculus. Unfortunately, no general method for the inverse pro-ess

of integration exists. It is necessary to derive a rule for the

integration of each function. The formulae of integration are

derived or disclosed by falling back upon our knowledge of the

rules of differentiation. In fact, the first simple rules, given in

Art. 18, are merely directions for retracing the steps taken in

differentiation. The inverse operation of finding an integral is,

in general, much more difficultthan the direct operation of find-ng

a differential or a derivative.

This chapter gives an exposition of the fundamental rules and

methods employed in integration. One or more of these rules

and methods will come into play in every case in which integra-ion

is required.

18. Fundamental integrals. Following is a list of fundamental

formulae of integration derived from the fundamental formulae

of differentiation. They can be verified by differentiation, as

36



17-18.] RULES AND METHODS OF INTEGRATION 37

indicated in the first of the set. An additional list is given in

Art. 22.

Every integrable form* is reducible to one or more of the

integrals given in these two lists. The student should have

ready command of these formulae for two reasons : first,so that

he may be able to integrate these forms immediately; and second,

so that he may know the forms at which to aim in reducing com-plicated

functions. The functions to be integrated will not

usually present themselves in terms of these simple, immediately

integrable expressions ; and therefore a considerable part of this

book is taken up with algebraic and trigonometric transforma-ions

showing how to reduce given functions to these forms.

In the following formulae, u denotes any function of a single

independent variable.

J n + 1+ 1

in which n has any constant value, excepting " 1. The case in

which n = " 1 is given in II.

Differentiation of each member of I: with respect to u gives

w* du,

n.t j^=logt"co = logu + logc=:logct".

The different ways in which the arbitrary constant of inte-ration

can appear in this form, may be noted.

* ** An integrable form " here means a function whose integral can be ex-pressed

in a finite form which involves only algebraic, trigonometric, inverse

trigonometric, exponential, and logarithmic functions.

t According to I., lu-^ du =

-^+ c = - + c=oo + c. Nevertheless,

JJ " 1 + 1 0

_lu-^ du can be derived directly by means of I. For, on putting f-cin + 1

for c, which is allowable by Art. 9, \u^ du =^^^ + Ci. Now ^"

QJ

n+1 n+1

= - when n = " 1. Evaluation of this indeterminate form by the method of0

the differential calculus gives, differentiating numerator and denominator

as to n, M'+Uogtt, that is, logw. Hence iu-^du=\ogu +ci.



38 INTEGRAL CALCULUS [Ch.III.

IT. Je~clw"~ + c.

y* rsin u dtt = " eoste 4 c"

VI. Jcos t*clt* = sin t* + c.

?!!" J8ecî"clutanu + c.

Tin. rcsc*f*clt*-cotM

+ c.

IX. J8eciitanf"cliisecf" + c.

X. Jescucotiiclu=-csciic.

XI. ('--^^ = sin-iM + c =

-cos-iM + ci.

XII. r-^^ = toii-iw + c =

-cot-iti+ ^

/

"ci.

Xm. f ^= = sec-it* + c =-csc-iM + ci.

Jtt Vm2 - 1

XIT. f^"

= Ters-i t* + c = - coTers"* w + ci.

Ex. 1. fx"da;=.^^+c=}x*c.

J^ J"5

+ 1"4

4x*

Ex.8. fjC2^"JE.= fâ+s^"^)=log(i+ sinx)+c.

J 1 + sinx J 1 + sinx

Ex. 4. fx*(?x,

fxi'^cix,

fs""+"d",

{t^dt,(z^^dz,p^dp.




EX. 5. j-. J'fIf,j-.- J-x-...-.-"^,g.

Ex. 6. fic*da;,

f"*d^ (x^dx,Jt?"dv, fa:"*(te,

fa;"*da;,

fx""(to.

Ex. 7. fVidaj,-^,f-^,fA^dx,f^cte, fVS?(to,r^.J J ^ J

y^J J J J J^

Jf' J"-l* Jie8+ 3' Juvw^-l' Ja"+ 2x2-2a; + 4'

tanx'

J sinx

Ex. 9. fc2"2(te, f*2*da;,(w + n)""to.

Ex. 10. fsin2xdf(2x),cos3x(f(3x), fseca4x(f(4x).

Ex. 11. fsecixtanixe^Cix),^^ . f~^^=, r_dOi5L

"^ "' VI-4x2 -^ VI - 9x-^ -^ Vl-tt2tttV

C 2dx C 3"to r 2dx r 4dx /* Û/^"^^ ^^- J 1 + 4x"' J 1 + 9x2' J

2xV4^^^rT'.J4xVl6^^^3l' J

x^^^^â2

19. Two universal formula of integration. In this article two

formulae of integration will be given which differ from those of

Art. 18 in that they do not apply to particular forms merely, but

are of a much more general character.

Suppose that /(a?),{x)y" (a?)," ", are any functions of x. Then,

A. j'{/(a5)l?'(a5)4"(a5)+-}"la5= (f(x)dx

-\-(jF(x)dx+(^(x)dx";

for differentiation of each member of A gives f(x)-\-F(x)

+"t"(x)\-

"-. This formula may be thus expressed : the integral

of the sum of any number offun"itionss equal to the sum of the

integralsof the several functions.



40 INTEGRAL CALCULUS [Ch. III.

Ex.1. \{x^"m\x"e')dx=\x^dx"\mixdx"\ef^dx^

=

Jas'C08X

" e* +c.

Each of the separate integrations requires an arbitrary con-stant

; but, since all these constants are connected by the signs

+ and "

,their algebraic sum is equivalent to a single constant.

Again, if u is any function of a, and m is a constant,

B* \mu dx = fn\u dx^

for differentiation of each member of B gives mu. Hence, a

constant factor can be removed from one side of the sign of in-egrat

to the other without affecting the value of the integral.

It will soon be found that sometimes it will make the work

simpler to remove such a factor from the right to the left of the

sign I, and that, at other times, the process of integration will

be aided by putting such a factor under the integration sign. It

follows from B, that the value of an integral is unaltered if a

constant is used as a multiplier on one side of the sign J, and as

a divisor on the other. Thus,

udz

m

\udx = " \ mu dx =

mi

This principle will often be found useful.

Ex.1. J3x"i"3j-x"i"=ij-2x"i"f." + c.

Note. The value of an integral is changed if an expression that contains

X is transferred from one side of the sign i to the other. Thus,

(x^dx=:iofic;

but x\ xdx = ^si^-\-c.

Ex.2. f7.r*dfa;,azÛz,(ad^ bxf^-^ dx, (a^Ûx, (^ ab^ cx^^-^ dx.




Ex.8. f(x2_2x4-6)da;,(a^ 3"* + 6

x*)da.

Ex.4. (iZ+ btydt, f(a*-x*)"*K"f(costf+ sin tf)W.

Ex. 5. fe" dx, (e^dx, Tet*x, (e'^'dx,ef^+^'+^x2)dx.

Ex. 6. f(cosmx + einS x) dx, I{sec x4- coseo*^ (m + n)"}dx,

Ex.7, f^" '^,f ^^^

,f ^^

.

Ex 8 f "^^ r cto r dy_

JWT^^' J 9 + 4x2' J 16 + 264Vl-^

20. Integration aided by a change of the independent variable.

Integration can often be facilitated by a convenient change of

the independent variable. For instance, if f(x)dx is not imme-iately

integrable, it may be possible to change the independent

variable from x to t, the relation between x and t being, say

X = ^{t)yO that f(x)dx is thereby put into a form F(t)dt which

can be easily integrated. Experience and practice afford the only

means of determining the substitutions that will be helpful in

particular cases. The actual substitution of the new variable

may often be conveniently omitted, as in Exs. 1, 2, 3, below.

Ex. 1. ('(xa)"dx.

On putting x-{- a=^t^ dx = dt\ and the given integral becomes

J n+1 n+1

Since dx = d(x + a),the given integral may also be written

f(x+

aYd(x

+

a),and x + a being regarded as the variable ti, the integral

is (^+ ^)"'''4-c,s before.

n+1



42 INTEGRAL CALCULUS [Ch.Ul.

Ex. 2. r8ec2(6-2a;)da;.

If 5 - 2a; = ", da; = -

Jd" ; and

fsec* (6 - 2x) da; = - l^B/^Hdt - Jtan " + c

= - Jtaii(6-2x)+c.

Since da; =

--Jd(6" 2

a;),heintegral may be written also

fsec2 (6 - 2x)da;= - J fseca(6 - 2a;)d(6- 2x)

=

-itan(6-2x)+c.

Ex. 8. fc"+""da;.

If a 4- fta; f, da; =

-dt^ and"

Ce"+"'da;-fcdf le"

+ c =i6"+""

+ c.

Otherwise : since da; =i d (a + "a;),"

f""+""da; =

i fe"+*'d (a + 6a;)

ie"+"" + c.

TT^ " C 12x2-4a;

+ 6^"^^"*-

J4xs-2x24-6x-10^-

On putting 4x8 - 2x2 + 6x - 10 = ", it follows that (12x2-

4x + 5)dx= dt^ and

Note, /if "/i"expression under the sign of integration is a fractionwhose

numerator is the differentialofth^ denominator^ the integral is the logarithm

of the denominator.

Ex. 5. f^-"^.J sin"*x

If sin X = {, cos xdx = d^ ; and

J sin^x J (^ bt^ Ssin^x

Otherwise :f"2!.*i5

^("'"")

=1_

+c.

J sin^x J (sinx)^ Ssin^x

The necessity of learning to recognize /orws readily will be apparent



20.] BULBS AND METHODS OF INTEGRATION 43

Ex. 6. f-^^-.

If x +

then ("^%= ^^^^^= f(--4 l-iW

If X-\-I

=: Zj dx = dz^ X = Z " If

Ex.7. f ?^(te.

If e* + 1 = 0, e*da; = d^;, e* = Z'-l;

andf_^_c?x=f-^:i!^= f(^lli)tf^=f(,i_;,-J)^

"^(e"4-l)* ''(6-+ l)i -^

."*

-^

= A("*+l)*(3e"-4).

f(2+ 3a;)*da;,f(3-7a;)*dx.

Ex. 9. \cos(x"\-a)dx,\aec\x+a)dx,\ "

^.^

^ .y\sm(a-\-hx)dx.

J J'

Jcos'*(4 O 05)

J

Ex.10. fcos^cZx,e2+""da;,i'^dx,f^^dx.

Ex. 11. fx(a + ")*d^,

f ^^^" *

"*

^*^ (a + 6x)*

Ex 12 f ^ rcos(loga;)da; r d^*

J (l + fl;2)an-la;'J a;

'

"^sin^f^'j.

Ex.18. {{a-^hz)'^dz,yJia^hxydx, {-r7=====:




Ex.14, f,

^" (Put2x-l=4jp.)

J Vl6 + 4a;-4a;"

Ex. 15. f22!9"^. (Put sin X = ".) f(sm*^ - 8 mn*e + 4 abcfleJ siii"x

^

+ 11 sin tf+ 2)cos$d$, f(tan"0 - 7 tan"0 + 2 tan^ + 9)sec"0d^.

2L Integration by parts. Two universal formulsB of integra-ion

were given in Art. 19. A third formula of this kind will

now be discussed. Differentiation

showsthat, u

and

v being any

functions of x,

d/ N

du.

dv

This may be written also in the differential form,

or more simply, d (uv) vdu + udv, (1)

in which, du

=-^dx,s^nd dv =

-^dx.x dx

Equation (1)becomes on transposition,

udv =

d(uv) vdu.

Integration of both members of this equation gives

"" \udv = uv " \v du.

Equation C may be used as a formula for integrating udv when

the integral of v du can be found. This method of integration,

commonly called" integration by parts," may be adopted when

f(x)dx is not immediately integrable, but can be resolved into

two factors, say u and dv, such that the integrals pf dv and v du

are easily obtained. The procedure is as follows :

if{x)dx= iudv]

whence by C, = wv " I v du.




No general rule can be given for choosing the factors u and dv.

Facility in using formula C can be obtained only by practice.

This formula has a greater importance and a wider application,

than any other in the integral calculus. The following examples

will show how it may be employed.

llogxdx = X log X " t X"

= xlogx " X4- c.

Ex. 8. Find fx"*dx.

Let tt = e", '^^'=

xdx ;

then, du = e'dx, " = J x^.

The formula gives

Cxe'dx JxV - J fxVdte.

But x^"i*dx is not so simple for integration as xe*dx. This indicates that

a different choice of factors should have been made.

On putting" m = x, dv = "*dx,

du = dx, " = e*,

and formula 0 now gives

\X"i'dx = xe'" le'dx

= xe* " e* + c




Ex. 4. Find (ufie'dx.

Let w = a^, dv = e'dx ; then du = Sx^dx, r = c*.

Hence, (ofie'x = ofie'- sCxê^x.

To find fxVda;,put u = a;^, ^v = e'dx; then du = 2x"te, " = e* ; and

(xê^dxaV - 2 Cxe'dx,

By Ex. 3, (xcdx= X6* - c*

-fc

Hence, combining the results,

(xê'dxe*(a!"

3x2 + 6x - 6) + c.

This is an example in which several successive operations of the same

kind are required in order to effect the integration. Many such examples

will be met, and usually a formula called " a formula of reduction "will be

found for integrating them. ** Integration by parts" is of great use in

deducing these formulae of reduction. In order to avoid making mistakes in

cases like Ex. 4, a good plan is to write down the successive steps in the

integration clearly, without putting in the intermediate work, which can be

kept in another place. Thus :

fx8e" "Zx = x8e" - 3 fxV dx

= x8e* - 3 [xV - 2 fx"* dx2

= x8c" - 3 [x2e" 2 (X6* e*)]4- c

=

e*(x8-3x2 4-6x-6)4-c.

Ex. 6. jsin~i X dx.

Ex. 6. I cot-i X dx.

Ex. 7. (za'z.

Ex. 8. (xâ'dx.

Ex. 9. Itan"i x dx.i



21-22.] BULES AND METHODS OF INTEGRATION 47

Ex. 16. fsec^ X log tAn xdx. Ex. 17. fsee"** dx,

Ex.16. fa;"(loga:)2(to. Ex.18. fx""logx(te.

22. Additional standard forms. Some fundamental integrals

which often appear are collected together in the following list.

Their derivation will be found in the next article.

XY. rtantt"lt"log8eeu+ C.

XVI. fcot udu = lo? sin w + C.

XVII. j'sectt"lwlogrtanf||Wc.

XVIII. Jcosee"It" = logrtan ^+ C

XX. f-/!i_=ltan-i^c =

-lcot-i^+C'.

XXIL f^=^^=:r =

yer8-i^+C-coyer8-i^+C".

XXIII. f_^*?L-=

-Llog ?^^=^+ C =

i-tonh-i+ C.Ji^2_"i2 2at"H-a a a

XXIV. f-"^^=: = log(tt+ "/fi2Ta2)+C =

sinh-i^+C'

XXV. f ^^= log (w + "/tt" a") + C = cosh-i

^+ C.



48 INTEGRAL CALCULUS [Ch.lU.

23. Derivation of the additional standard fonns.

Formula XV. Since tan u = îBif,cosu

J J cosw"/

cost*

= " log COS u = log sec u,

Fonnula XVI. fcotdu= C^^du =C^Sm^

J J suiu "/ sinu

= log sin 1* = " log cosec u.

Formula XVIII. Since cosec u = cosec u^Qsecu-cotu

cosec tt " cot It

/"."""/", ^-.^ r" cosec tt cot tt + cosec* u ,

,

cosec uati= I ' duJ cosec tt "

cottt

/d(cosec " cot u)cosec M " cot u

= log (cosec " cot u)= log

cosec u " cot tt

1"cost*

sinu

"

28in"|

2sm^co8^^

Formula XVII. On substituting w +|for w in XVIII., there

results

J*cosecu

+^du

log

tan^|4-|\'

that is, Isec udu = log tan ( 4-j )

Formula XIX. If u = aay then du = adz, and

"

"1

*"

-1

*"

= Sin*

- = " cos*

"



23.] RULES AND METHODS OF INTEGRATION 49

Formula XX. If w = az, then du = adz, and

1,,tt

1.

,tt= - tan~^ - = cot"^ "

a a a a

Formulae XXI., XXII. These integrals also can be derived by

means of the substitution u=

az.

Formula XXIII. Since"1"

=

-^(" -^\" "r 2a\u " a u + aj

J u* " 0? 2 aJ \u " a u + aj

= A Uog (w -

a)- log (w+ a)}

2a u + a

Formula XXIV. If u^-\-a^ = s?, then udu " zdz,

du__

dz

z u

Hence,

and, by composition.u-\-z

Therefore, f^^

=f^?^

"/ Vtt^+ a^J t* + z

= log (tt 2;) c = log (u+ V w* + a*)+ c

= log!i"^^"^ + c. = 8inh-"'i+ c.



50 INTEGHAL CALCULUS [Ch. IIL

The latter result can be derived also in the following way :

On putting u = az,

r_du_^ r-^= sinh-^ + c

= sinh-* - + c

a

Formula XXV. Similarly, on

puttingi** " a* = a*,

= log h c, = cosh"* - + (^

Or,putting

" = cw;,

f /^ = r-^==cosh-^4-c = cosh-*!*+ c

Ex. 1. Find fVaa-x^da;.

Integrating by parts, let

tt = Va2 " "'", dv = dx.

Then dti =

^

(Zi;, v = x,

andfVgg - a;g(fa= xVg^ - x^ + f

"

^Jg~.

.

Since VâZT^ =

^^ "" ^^

,it follows that " ^^ =

^^- Va^- x^.

Vo2Z^ Va2-x2 Va*^-x2

Hence, fVa^ - x^dx = xVa" - x^ 4- a^ f ^ f Va^ - x^dx.

From this, on transposing the last integral to the firstmember,

fVa-2-x2(toi /xVa2 - x^ + a^sin-i ^V



23.] RULES AND METHODS OF INTEGRATION 61

Ex. 3. r ^ /^ f ^

=

if^

=sin-i f^-^-^V

jj^ 3r dx

_

r dx

"'Vxa4-8x4- 62JV(x 4- 4)"

-f36

= log (X 4- 4 4-Vx2 4-8x4- 62).

Ex. 4. f ^^=f

^ =

-JLtan-i^+^.x2 4-6"4-12 J (X 4- 3)24- 3 V3 \/3

3)~2Ex. 5. f ^

=f^

=llogI^"

Jx*4-6x4-6 J(a;4-3)"-4 4'^

(x 4- 3)4-2

=hog^"i.4 *x4-6

Ex. 6. f ^" Ex, 15. f-^-.

"^ \/x2-6x^ a 4- x"

\/x2-6x

y/fdxx. 7. f ^^^.

Ex.16, f-^.^ V7x*4-19

J y" - 8

Ex. 8. f

^^^Êx.17,

f-^^."^ V3x*4-2x2-l

^ tan ox

Ex. 9. f "

Ex, 18. fcot (ax 4- b)dx.

"^ v4x " x^

Ex 11 f ^ Ex, 80. f^

r= f ^(^-2) ].JV43^* - x-^-4x4-8L

J(x-2)H4j

Ft 12 f_^?_.

Ex, 21. f^

^

^'^^^'

JVT^Jxa-4x-8

Ex. 18. f6tan3x(te. Ex. 88. J(sec2 x 4- 1)""te.

Ex, 14. f-4^=- Ex.83, f ^

"^\/3-6x3 "'(x--a)V(x-a)2~6"INTEGRAL CALC. " 6




Ex. 24. f Êx. 89. f ^

Ex. 26. f-^^. Ex. 80. (l"^^de,"^ V x* " c* J sin ^

Ex. 86. ^xyV^fi^dx.^^ ^^

r dx

^yJaH^ + 2 6a;+ c

Ex.87, f ^Jl

Ex.88, f ^

^ Vl6 X " ttx'* [Rationalizehe numerator.]

24. Integration of a total differential. It has been shown in the

differential calculus, that if

"=/(".y). (1)

Xy y, being independent variables, the total differential of u is

equal to the sum of its partial differentials with respect to a; and

y. That is,

(ltt ^d"+ ^dy. (2)ox ay

It will be remembered that when differentiation is performed

with respect to a?, y is regarded as constant, and when differenti-tion

is performed with respect to y, x is regarded as constant.

Suppose that a differential with respect to two independent

variables is given, namely,

Pcbi+Qdy, (3)

in which P and Q are functions of x and y. The anti-differential

of (3)is required. Not every function (3)that may be written

at random has an anti-differential. Hence, it is necessary to de-ermine

whether an anti-differential of (3)exists or not, before

trying to find it. It has been shown in the differential calculus

that if u and its first and second partial derivatives with regard

to Xy y are continuous functions of x, y, then.

dy dx dx dy(4)




If (3)has an integral, say w, then,

du "

Pdx-\-Qdyj (5)

in which P=$^, (6)ox

and Q = g. (7)

Differentiationof

both

members of (6)and (7)with respectto

y and x, respectively, gives

dP dû

dy dy dx

dx dx dy

Hence, by (4), |=^. (8)

Therefore, if (3)has an integral, relation (8)holds between the

coefficientsP, Q,and the differential (3)is then said to be an exact

differentialConversely, it can be shown that if relation (8)

holds, the differential (3)has an integral. For the present the

latter proposition may be assumed to be true.* The condition

(8) is called the criterion of the integrability of the differen-ial

(3).

Suppose that the coefficients P, Q satisfy the test (8),then

there is a function u which satisfiesequation (5). Since Pdx can

have been derived only from the terms that contain a?,integration

of the second member of (5)with respect to x gives

fPdx +c.

in which c denotes any expression not involving x.

"For proof, see Introductory Course in Differentialquations, Art. 12,

by D. A. Murray (Longmans, Green " Co.).



54 INTEGRAL CALCULUS [Ch.111.

Now Q dy has been derived from all the terms of u that contain

y. Some of these terms may contain x also ; and if so, they have

been discovered already in jPdx, Therefore, the remaining

terms of u that do not contain x will be found by integrating

with respect to y the terms of Qdy that do not contain x. Hence,

the following rule: Integrate P da? as if y were constant; inte-rate,

as if a? were constant, the terms in Qdy that do not contain

X

; add the results andthe

arbitrary constant ofintegration.

Ex. 1. Integrate ydx-{-xdy,

dP dOHere P = y, Q = x; hence ^ = 1*

-^= 1" and thus, criterion (8) is

satisfied.

Also, iPdx = iydx = xy; and there are not any terms in Q dy without x.

Hence the integral isxy + c.

Ex. 2. Integrate ydx " xdy,

riPdO

Here P=y, Q= "x;hence " = 1,

-^= " 1, and the criterion is not

satisfied. Therefore an integral of the given expression does not exist.

Ex. 8. Integrate (x^ ^xy - 2^) dx-\-(^

- ^xy - 2x2) dy.

Ex. 4. Integrate (a^-2xy-y^)

dx- (x + y)^dy,

Ex. 5. Integrate (2 ax-\-

by"\-g)

dx"\-(2ay + bx + e) dy,

25. Summary. The directions so far given for obtaining the

indefinite integral of f(x)dx may be summarized as follows :

(1) Memorize the fundamental formulae of integration, given

in Arts. 18 and 22.

(2) Acquire familiarity with the application of the principle of

substitution, or change of the independent variable, discussed in

Art. 20.

(3) Use the firstand second universal formulae of integration,

A, By given in Art. 19.

(4) Learn to apply with ease the third universal formula of

integration, namely, the formula for integration by parts given in

Art. 21.






10. f "f^

+.

^ Va;,n(sm2^ + cos30)d^,

J""toLVa;2 + a2 - Vi^^âJ

11- f:===" (Pat a; = -. Compare result with formula XXI)

f

_^.

(Put sin a; = "f. Compare with XVII.)

12. f ^(Put3+2a;=a?.)f

-^-. (Putl+a;="2.)

JV1-3X-X2

"^(i+a;)f+(i+^)J

18. rî^l^dir.^^ . (Puta + 6x2 =

;52.)*^ 2 "" "^ Va + 6x2

14. Jsec-ixclx,cosec~ix(te,cos-ix"lx,x2sin""ix(te.

16. Ixcosxdx, Jx^sinxdx,xtan2xdx.

16. fx'a'dx, Tsui log cos X(ix, rcosec2xlogcotx(lx.

17. f(2x-2)(x2-2x+ 6)cos(x2-2x + 5)fZx. (Putx2 -2x + 6=;?.)

18. fx61og(x"+ a8)dx. (Putx8 + a8 = ".)

19. I ^ ^^ dx, (Integratey parts, putting u = (logx)2.)"^

x^

20. Show that

f(logx)"dx=

x[(logx)"-

m(log x)"-i+ m(m" l)(logx)""-* "

+ (- 1)""im(m- i)...3.21ogx+ (-l)".ml].

21. Show that

fe"x" dte= x"e" - m fe*x"-i dx =

6*[x* mx"" -1 + m(m" l)x""-" """

+ (-_l)"-%(m-l)...3.2.x + (-l)"".wl].



25.] RULES AND METHODS OF INTEGRATION

22. r(sec + cosec xy dx.

28. f-i*-J 1 + COot^

dx24

-x^

r dx

J Va2 + 62 _-

25. Evaluate 2 f^log

tan 0 . cosec 2ede + log cot ^f cosec 2 ^ "W.

26. f(a

tan ^sec ^+ 6) cot ^"W.

"^V2 + 2a;-"2 J V- 16 -

28. flog("+Vx2-a2)^

dx_

lOx-i

(2x

29. flog(" + VS^Tl?) "^.

30. jJ'^^^de.

g-rasecgg + ftcosec^g^p

3gfsecxtanxda;

* J tan^ + cot^

* '

J tan2x-2

oo C 1^ r. 39. f" ^.2. jvers-i -'Vxdx. J e" " 6 e"*

^ ^ 40 f "^^

88. jV"Ta^dx,'

Jy/a^-b

r ^^M^ r sinxdx

"^xi-5xiJsm-Vcosx

85. f^-^ 42. f '-

Jsinx+.cosx

.Jax2 +

2

dx

6x + c

36. f,

^^.

" 48. f ^

37.r(3x + 4)V^+T^^^ 44.

r

c

dx

V2 " X. V- ax2 + 6x + c

45. Integrate sin x cos y dx + cos x sin y dy.

46. Integrate cos x cos y dx " sin x sin y dy.

47. Integrate (3x2 + 6xy + 4y2)dx +(3x2 + Sx^^ + 6y2)dy.

67



CHAPTER IV

GEOMETRICAL APPLICATIONS OF THE CALCULUS

26. Applications of the calculus. In this chapter some of the

practical applications of the integral calculus are discussed. In

particular, the areas of curves and the volumes of solids of

revolution are determined. Art. 32 deals with the deduction

of the equation of curves from data whose expression requires

the use of differential coefficients.

There is one common

aimin by far the larger

number of the

simpler applications of the integral calculus. This aim is to

find the sum of an infinitenumber of infinitely small quantities.

The process of summation has been discussed in Chapter I.

The student will find that in most of the problems there are

two steps to be taken in order to obtain the solutions, viz.:

(a)To find the

expression

for

any

one

ofthe infinitesimal

quantities concerned and to reduce it to a form that involves

only a single variable;

(b)To integrate this differential expression between certain

limits which are assigned or are determinable.

Each of the differential expressions is called an element, "

an element of area, an element of length, an element of volume,

an element of force, etc., as the case may be.

27. Areas of curves, rectangular coordinates. It has been

shown in Arts. 3-5 that the area* between the curve y=f(x),

* The calculation of such an area iscalled** Quadrature of curves." From

this comes the phrase ^^to perform the quadrature," which is often used as

synonymous with **to integrate." The areas of only a few curves could be

found before the discovery of the calculus. Giles Persone de Eoberval (1602-

68



26-27.] GEOMETRICAL APPLICATIONS 59

the oâxis, and two ordinates for which x = ay xssb, is ex-pressed

by

f/(x)dx.It has also been shown that this area can be evaluated by

finding the indefinite integral

of f{x)dx,substituting b and a

in turn for x in the indefinite

integral, and taking the differ-ncebetween the results of the

two substitutions.

Ex. 1. Find the area bounded

by the parabola whose equation is

y^ = 4 ox, the axis of x, and the

oi*dinate at a; = xi. Also find the

area between the parabola y^ = 9x^

the axis of x, and the ordinates for

which " = 4, jc = 9.

Let QOC he the parabola whose equation is y* = 4ax. Take 0M= xi,

0A = 4, CD = 9; erect the ordinates MP, AB, DC, Suppose that two

ordinates B8, VT are drawn at a distance dx apart. The element of area,

which is the area of any infinitesimal rectangle like BSTV, \b ydx. The

area required in the first case is equal to the sum of the areas of all such

rectangles, infinite in number, that are between OY and PM; that is, be-ween

the limits zero and xi for x. Hence,

Fig. 16.

area of 0PM = [\dx.First of all, y must be expressed in terms of x. This can be done by

means of the equation of the curve, from which

y = " 2a^xi

1676), professor of mathematics at the College of France in Paris, Blaise

Pascal (1623-1662),ohn WalUs (1616-1703),avilian professor of geometry

at Oxford, considered an area to be made up of infinitely small rectangles,

and applied the principle to the determination of the areas of parabolic

curves. The French geometers iound the formula for the area between the

curve

y= x", the

axis ofx,

and any ordinatex = h

whenm is a

positiveinteger. Wallis found the area when m is negative or fractional. This was

before the development of the calculus by Leibniz and Newton.



60 INTEGRAL CALCULUS [Ch. IV.

Hence,

that is,

(The positive sign denotes an ordinate above the x-axis, the negative, one

below.)

area of 0PM =

("^2aMdx

area of 0PM = two thirds of the area of the circumscrib-ng

rectangle OLPM,

The area OPQ = 2 0PM = two thirds the area of the rectangle LPQR,

In the second case :

area-4J5C2)=f

ydx

= 3 ("Veto3 [f"* + c]9= 38.

If the unit of length is an inch, the area of ABCD is 38 square inches.

Ex. 2. Find the area between the curve 2/^= 4 ax, the axis of y, and the

line whose equation is 2/= 6.

X

_A_

B

Fio. 16.

In this case it is more convenient to take for the element of area the

infinitesimal rectangle indicated in the figure. The element of area is thus

xdy\ and

area OAB=i \ xdyJo

Joâ

12 a



27,] GEOMETRICAL APPLICATIONS 61

Ex. 8. Find the area of the ellipse -5 + 7^= !"

The area required isfour times the

area of the quadrant A OB. An ele-ent

of area is the area of an infini-esimal

rectangle BSTV, namely

y dx. The sum of allthese elements

from O to ^ isexpressed by \ ydx,Jo

From the given equation,

Fig. 17.

a

in which the positive sign denotes

an ordinate above the x-axis, and

the negative sign, an ordinate be-ow.

Hence,

area of ellipse= 4 OAB = 4 \ydxJo

aJQ

which by Ex. 1, Art. 23, = ^ [|S^^^â + fsin-ic]*= Tab,

If a = 6, the ellipse is a circle whose area is ira^.

Find the area included between the ordinates for which " = 1, a; = 4, the

curve, and the axis of x.

Area PQMN= ^^ydx â^ - x^ dx

="rEVS2T:^

+ ^sin-i^cT12 2 a Ji

= 5|2Va2_l6iVoaTTT + ^fsin-i^sin-iiÎf the semi-axes are 5 and 3,

area PQMN = f{6 - V6 + " (si^"^ - si^"^ *)"

= f{6- 2.454)+ ^ (.927.201)

= 3.778.




Fxo. 1"

Ex. 4. Find the area between the

curve whose equation is

y =

A(aJ-l)(aJ-3)(x-6),

the axis of x, and the ordinates for

which X = " 2, a; = 7.

7

The area required z^\ydx

"27

-

^C(pfi

9a;2 + 23"-

16)"to

Further remark on this example

may be instructive. On putting

y=

0, the intersections of the curve

and the x-axis are seen to be at the

points for which x = 1, 3, 5. That

is, referring to the figure, OC = 1,

0Z" = 3, 0E = 6.

1-i

Area ^P(7= fydx = - W-

This area appears with a negative sign, since the ordinates are negative in

APC because itis below the x-axis.

Area CHD = Ty dx = + J,

the sign coming out positive since CHD is above the x-axi"

ATea,DLE=(ydx =

-\;

and.

area EQB = \ ydx = + S.

The area required = area APC + area CHD + area DLE + area EQB

The absolute area=W

+ i + i + 3 = 12^.

as obtained before.




This is an example of the principle indicated in Art. 5, namely, that when

the area between a curve, the x-axis, and any two ordinates, is found by inte-ration,

this area is really the sum of component areas, those above the

X-axis being affected with the positive sign, and those below the x-axis with

the negative sign. The next example will also serve to illustrate this.

Ex. 5. Find the area between a semi-undulation of the curve y = sin x

and the x-axis.

The curve crosses the x-axis at x = 0, x = ir, x = 2 ir, etc.

Area of ABC = \y dx = f'^sinx dte = [- cos x + c]'= 2.

Jo Jo "

But BTea,ABCDE=C''ydx=C''6mxdx 0.Jo Jo

The total area, regardless of sign, is 4.

Y

Fie. 19.

28l Precautions to be taken in finding areas by integration. The

method offinding

areas whichhas been described in

the last

article can be used immediately and with full confidence in the

case of a curve y=zf(x),only when the limits a and b are finite,

and the function f(x)is continuous and one-valued for values of

X between a and b, and does not become infinite for any value

of X between a and b. Special care must be taken in cases in

which anyone

ofthe

conditions justmentioneddoes

nothold.

While, in some of these cases, the application of the method of

Art. 27 will give true results, in other cases it will give results

that are altogether erroneous. A few examples are given below

in order to emphasize the necessity of caution.

Ex. 1. There is a double value for y in the parabola y^ = 4 ax. This was

considered in Ex. 1, Art. 27.

Ex. 2. Find the area included between the parabola (y " x " 6)2= x,

the axes of coordinates, and the line x = 6.




In this case y = " " Vx + o ; and thus to each value of x belongs two

values of y. The ambiguity can be removed by defining more exactly what

area is meant. If the area OBPM is desired, the value of y corresponding

to each value of x between 0 and 5 is x " Vx + 6. Hence,

area OBPM= (\x - Vx + 6)dx

V M

Fio. 20.

If the area OBQM is desired, the value

of y corresponding to each value of x be-ween

0 and 6 is x + Vx + 5 ; and hence,

areaOjBQJIf=f% + Vx + 6)dx"/o

=

V- + ""/^.

If the area PBQ^ between the curve and the line x = 6, had been required,

it would have been necessary firstto determine the areas OBPM, OBQM.

Area PBQ = area OBQM- area OBPM;

= ^V5.

Another way of finding the area of PBQ is the following. Let TS be any

infinitesimal strip of width dx parallel to the y-axis. Evidently, TS is the

difference of the values of y that correspond to x= OV. Hence, denoting

these values of y by yi, ^2,

area PBQ =\(yi- y^)dx

Jo

= 1{(x+ Vx + 6)-(x-Vx + 6)}dx

= r 2Vxdx

= ^V6.

Ex. 8. Find the area included between the witch y =

o"

and its

x2 + a2

asymptote. The asymptote is the axis of x, and hence, the limits of integra-ion

are

+oo

and" oo. In this case it is

allowableto use infinite limits.

Eor, on finding the area OPQM between the curve, the axes, and an ordi-ate

at distance x from the origin,



28.] GEOMETRICAL APPLICATIONS 66

area OPQM= CydxJo

Jo "2 + a2

rrfanan-i^H-cTa Jo

= a2tan-i^.a

Fig. 21.

If the ordinate MQ be made to move away from the origin towards the

right, that is, if the upper limit x increases continuously, then tan-i-

increases continuously, and approaches - as a limit. Hence,

2

r^0

o

x"^ + a^

ira-*

represents the true value of the area to the right of the y-axis. Since the

curve is symmetrical with respect to the ^-axis, the area required is double

this, namely, va^,

Ex. 4. Find the area included between the curve y* (x^"

a^y= 8sc*,the

a;-axis, and the asymptote x = a.

2x

In this case y = "

To every value of x corresponds a real value

of y ; but, when " = a, y is infinite. Therefore a special examination is re-quired.

For values of x less than a, however, y is finite. Then, for a: " a,

area OMP2xdx

= (




As X approaches the value a, it is apparent that the area OMP approaches

3 a' as a limit ; and hence,

.

Fio. 93.

Ex. 5. Find the area bounded by the curve in Ex. 4, the x-axis, and the

ordinate at x = 3 a,

It has already been noticed in Ex. 4, that /(") becomes infinite when

x = a. Asa lies between the limits of integration, 0 and 3 a, the integration

formula for the area should not be used until its applicability is determined

by a special investigation. The area from x = 0 to the infinite ordinate at

X = a, has been shown in Ex. 4 to be 3 a'. The area to the right of the

ordinate at x = a will now be discussed. Since /(x) is finite for values of x

greater than a, then for limits, x " a and x = 3 a,

area Jf'P'CiVirf-2xdx

î

= 6a*-3(x2-o2)*.

As X diminishes and approaches a, this area approaches 6 a* ; and hence,

the area between the infinite ordinate at x = a, and the ordinate at x = 3 a,

is 6ai.

Hence, the total area between the curve, the x-axis, and the ordi-ate

atX = 3a, is 3 a"

+6 a", that is Da'.

The same result is obtained when I " is evaluated in the ordi-




nary way ; and thus, the integration formula for the area holds good in this

case, although /(x) becomes infinite for a value of x between the limits of

integration.

Ex. 6. Find the area included between the curve y(x"

a)^= 1, the axes,

and the ordinate x = 2a.

Immediate application of the integration formula gives for the area,

r^ dx_

r 1_

_,.gl2"_

_

2

Jo (X -

a)2L X - a Jo a

FiQ. 28.

But, /(x), which is the length of the ordinate y, becomes infinite for

x = a; and, if an investigation be made similar to that carried out in Exs.

4, 5, it will be found that the area is infinite. For, OM being equal to x,

area OMPB= C ^, =

-iL

Jo (x "

a)''â" x a

It is evident, that as x increases from 0 to a, the area increases from 0

to CO. Consequently, the area between the curve, the axes, and the ordinate

at X = a is infinite. Similarly, it may be shown that the area between the

curve, the x-axis, and the ordinates at x = a, x = 2 a, is infinite. Therefore

the total area required is infinite. Hence, the integration formula for the/"2a

f[xarea, namely, i ? fails in this particular case in which f(x) be-vo (x-a)2

^*"

^ ^

comes infinitefor a value of x between the limits of integration. This con-clusion

may be compared with that in Ex. 5.

29. Precautions to be taken in evaluating definite integrals. It

has been shown in Art. 6, that any definite integral, say jf{x)dXj

maybe

graphically representedby the area between the curve

y =/(aj),he axis of aj, and the ordinates at a? = a, a? = 6. Hence,

INTEGRAL CALC. " 6






30. Volumes of solids of revolution. Let PQ be an arc of a

curve whose equation is y =f(x).Draw the ordinates AP, BQ,

and let OA = a, OB=b, The vohime of the solid PQML

generated by the revolution of APQB about the a;-axis is re-quired.

On the revolution of FQ each point in the-arc PQ

will describe a circle. Suppose that AB is divided into n equal

parts Ax, and let OQi=a?, QiQ2 = Ax, Construct the rectangles

^1^2? P2Q1 as indicated in the

figure, and suppose that they

have revolved about OX with

APQB,

It is evident that the volume

of each plate, such as PiPj-^g-û

of the solid of revolution is less

than the volume of thecorre-

sponding

exterior cylinder gen-

.eratedby the revolution of the

rectangle Q1R1P2Q2 about the

aj-axis,and greater than the vol-me

of the corresponding interior cylinder generated by the

revolution of

the

rectangleQ1P1R2Q2 about the aj-axis. Now,

the volume of the cylinder generated by Q1P1R2Q2

Fig. 24.

=

ttFiQiAaj

and the volume of the cylinder generated by Q1R1P2Q2

=

TrP2Q2Ax

=

7r[/(ajAaj)PAaj.

Hence, w [f(x)YAx" PiP2^2î " ^ [/(x+ Ax)yAx,

Suppose that PQML is divided into n plates, such as P1P2N2N1,

one plate corresponding to each segment Aa; of ^B; and suppose




also that the interior and exterior cylinders corresponding to

each of these plates are constructed. Then, on taking the sum

of all the interior cylinders, and the sum of all the exterior

cylinders, and the sum of all the plates P^P2N2Ni, the latter

sum being the volume required,

VTT [/(a?)]2Aa;PQML " V ^ [/(^+ Aaj)]2Aa?.

X = a 36 = 0

As Aa? approaches zero, the sum of the exterior cylinders ap-roache

equality to the sum of the interior cylinders. The

difference between these sums is at the most an infinitesimal of

the first order when Aa? is an infinitesimal, and accordingly has

zero as its limit. Therefore, since the volume required always

lies between these sums,

a;=6

volume PQJfL = limit^

V"^[/(")?Aa;;

that is, volume PQML = (\ [/(a?)] dx.

The element of volume is tt [/(a;)]x ; this is usually written

Try^dx, since y =f(x).This value of the element may readily be

deduced from the figure on supposing that Q1Q2 is an infinitesimal

distance.

If an arc of y =f{x)between the points for which y = c and

y = d revolves about the y-Sixis, it

can be shown in a similar way that

the element of volume is wo^dy, and

that the total volume generated by

the revolution is

TTj^^^y-

Before integrating it will be neces-sary

to express x^ in terms of y.



30.] GEOMETRICAL APPLICATIONS 71

E: '1. Find the volume of the right cone generated by revolving about

the X: 'is the line joiningthe origin and

the poiii (Ji^).Let M be the point (A,a).

The equation

of OiJf is

ax = hy.

The elemt it of volume iswy^

dx. Hence,

velum i OMN = TT I 3

Jo h^

y'^dxI

3Fio. 26.

This may be interpreted : the volume of the right circular cone QMN is

equal to one thi 'ithe area of the base by its altitude.

Ex. 2. Find 'he volume of the cone generated by the same line on revolv-ng

about the y-^xis.In this case, he ejement of volume

is TTX^ dy. Heni 3,

volume OM V= ir 4 x'

^0

Jo a'2

3

Ex. 8. Find the ^ olume of the solid

generated by revolv.ng

the arc of the Fig. 27.

parabola y^ = 4px t :tween the origin and the point for which x = xu about

the X-axis.

In this case,

volume OPPi = ir ('\^dxJo

/"Xl

= IT I 4pxdx

Jo

= 2irpxi^;

or, since yi^ = 4pxi,

Hence, the volume is one half the

volume of the circumscribing cylinder.



72 IXTEGUAL CALCULUS [CH.r*;V.

Ex. 4. Find the volume generated by revolving the arc in Ex. 3 alx * (^xxtthe

y-axis.^

In this caê,,(

volume OPQ =

ttCxî ^ly

f" "

, y*

or, since yi^=

4pz'-.

volume OPQ = ^ t \jfiXi^,of the 't oy

ence, the volume required is one fifththe volume of the '^"oyUnder of baae

PQ and height CO. ^

JiEx. 6. Find the volume of the solid generated by the rev f

olution about the

X-axis of the arc of the curve y = (x + l)(x+ 2) between t'che points whose

abscissas are 1, 2. L

Ex. 6. Find the volume of the cone generated by the revc 'jiution about the

X-axis of the parts of each of the following lines intercepted Ibetween the axes "

(a)2x + y = 10;^

(c) 4x - 5j/ + 3 =

-S0 ;

(6) 7x + 2y4-3 = 0; (d)3x-8y = 5. '

Ex. 7. Find the volume of the cone generated by the r /evolution about the

y-axis of the parts of each of the following lines intercepte fd between the axes "

(a) 4x + 3y = 6; (c) 5x - 7 y + 35

r|=;

(ft)3 X - 4 y = 6; ((Z)2 x + 6 y + 9 = )0.

*

Ex. 8. Find the volume of revolution about the x-axi % of the arcs of the

following curves between the assigned limits :

(a) y2 = x3, X = 0, X = 2 ; (6) (a^+ x'^)*= a ^ x = 0, x = a.

Ex. 9. Find the volume of the solid generated by tVierevolution about the

X-axis of the curve y^ = ex from the origin to the poir/those abscissa is Xi.

Ex. 10. Find the volume of the solid generated *by the revolution of the

same arc as in Ex. 9 about the y-axis.

Ex. 11. Find the volume of the solid generated by the revolution about

the y-axis of an arc of the curve in Ex. 9 from the origin to y = y^.

Ex. 12. Find the volume of the prolate spheroid generated by the revolu-ion

oi the ellipse " + ^ = 1 about the x-axis.




31. On the graphical representation of a definite integral. In

Arts. 4, 6, attention has been drawn to the principle that any

definite integral, whether it denotes volume, length, surface,

force, mass, work, etc., may be graphically represented by an

area.* A simple illustration may put this in a clearer light.

Ex. Find the volume of the right cone generated by revolving about the

jc-axisthe line drawn from the origin to the point (4,1).

Let P be the point (4,1),

andlet

POQbe the cone

ofrevolution. The equation of

OP is 4 2/= ".

Hence,

vol. P0"= fVy2(fx

The volume isthus | ir cubic

units of the same kind as the

linear unit employed. In

order to represent this volume graphically, draw the curve OHH whose

equation is

" x^ being the function of x under the sign of integration in (1);and draw

16

the ordinate CB at " = 4. The area BOG graphically represents the volume

.POQ.For,

area ^0C= ( ydx

Jo 16

Equations (1) and (2) show that the number of cubic units which indi-ates

the volume of POQ is the same as the number of square units which

indicates the area of BOG. In the same way, if the ordinate NH be drawn

at any point iV, for which x = a, say, it can be shown that:f\

ra^ denotes

both the number of cubic units in the cone MOL and the number of square

=r

"dx (2)

* On account of this property the process of integration was called by

Newton and the earlier writers*' the method of quadratures."




units in the area HON. It is thus apparent that the number of cubic units

in LMPQ is the same as the number of square units in NHBC, namely

^^(Q^-a^). Hence,

vol. POQ : vol. MOL : vol. LMPQ

= area BOG: area HON : area NHB C. (3)

If the curve y"^

rmrx^ be drawn, the numbers which indicate the areas

will be wi times the numbers which indicate the volumes of the correspond-ng

sections of the cone. But the ratio of any two right sections of the cone

will be the same as the ratio of the two corresponding areas, and proportion

(3)will stillhold. The curve 2/=

A wi^^'*^ can therefore be used to represent

the volume. It is sometimes well to use a multiplier m for the sake of con-venience

in plotting the curve that will graphically represent the integral.

Note. If the firstintegral curve (seeArt. 15) of OHB, namely,

JoX2

_

_L^3-3

'o 16 48'

be drawn, its ordinates represent both the areas of the segments of OHB

and the volumes of the segments of the cone POQ measured from O.

32. Derivation of the equations of certain curves. Oftentimes,

when a curve is described by some property belonging to it,the

formal analytic statement of the property involves differential

coefficients. In these cases the derivation of the equation of the

curve

consistsin finding a

relationbetween

the coordinates which

will be free from differentials. Examples of this have been given

in Art. 12. A few additional simple instances are introduced

here. In the larger number of cases the derivation of the equa-ion

of the curve will require a greater knowledge of differential

equations than the student possesses at this stage; and hence

further

problems ofthis kind

will

be deferred

until

Chap. XIII.

Ex. 1. Determine the curve whose subtangent is n times the abscissa of

the point of contact. Find the particular curve which passes through the

point (5,4).

Let (",y)be any point on the curve. The subtangent is y ". By the

given condition,^

dx

y'^=nx.ay

This may be written, " =^.

X y




Integrating, log c + log x = n log y ;

whence, y" = ex. (1)

All the curves obtained by varying c, satisfy the given condition.

If one of the curves passes through the point (6,4),for instance,

4" = 6 c. (2)

Substitution in (1)of the value of c from (2)gives

5y" = 4"x,

as the equation of the particular curve through (6,4).

What curves have the given property forw = l? n = 2? n = }? w = i?

Ex. 2. Find the ctirves in which the polar subnormal is proportional to

(isk times) the sine of the vectorial angle. What particular curves paas

through the point (0,2 tt)?dr

The polar subnormal is "

. By the given condition,dd "

^= ifcsin ^.

de

Integrating, r = c " ifcos ^.

For the curve that passes through (0, 2ir), 0 = c " k; whence e = k.

Hence, the equation of the particular curve required is

r = A;(l cos^),

.theequation of the cardioid.

Ex. 3. Determine the curve in which the subtangent is n times the sub-ormal

; and find the particular curve that passes through the point (2,3).

Ex. 4. Determine the curve in which the length of the subnormal is pro-ortional

tothe square of the ordinate.

Ex. 6. Determine the curve in which the subnormal is proportional to (is

k times)the nth power of the abscissa.

Ex. 6. Find the curve in which, for any point, the length of the polar

subtangent is proportional to (isk times)the length of the radius vector.

Ex. 7. Find the curve in which the angle between the radius vector and

the tangent at any point is n times the vectorial angle. What is the curve

when w = 1 ? when n = i?




EXAMPLES ON CHAPTER IV

1. Find the area of the figure bounded by the curve ac* +aafi

+a^

+ 6^ = 0, the "-azis, and the ordinates at as = 0, x = a.

2. Find the area inclosed by the curve scj^* y*+ 2 y* + 6 and the lines

a; = 0, y = 0, y = 1.

8. Find the area included between the parabolas y^ = 4 ox and x^ = 4 ay.

X X

4. Find the area included between the catenary y =

-(e*+ c *), the

axes of coordinates, and the line x = c,

5. In the logarithmic curve y = "f* prove that the area between the

curve, the axis of x, and any two ordinates is proportional to the difference

between the ordinates.

6. Find the area included between the curve y = " - "

* and the line

7. Find the 'area bounded by the curve y = x^-{-ax^, the ac-axis,and

(a) the ordinates at a = " a, and x = 0 ;

(6) the ordinates at x = 0, x = a.

8. Find the area inclosed by the axis of x, and the curve y = x " x*.

9. Find the entire area of the curve y^ = a^x^ " x*.

10. Find the area included between the curve

y^ (a* x^)=

a^* andits

asymptote x = a.

11. Find the entire area contained between the curve y^ (a^ x^)= a*

and itsasymptotes x = a, x = " a.

12. Find the area included by the curve x^y^ (x^ a^)= a" and its asymp-ote

X = a.

18. Find the area

ofthe loop

of

the curve

a^^= x*

(6+

x).

14. Find the total area bounded by the curve a^y^ + b^ = a'^Hhi^.

15. Find the volume of the solid generated by the revolution about th"^

X-axis of :

(a) y2"

ajS _ a;2 between the ordinates x = 1, x = 2 ;

(6) (a^ x)2y*= a^ between the curve and its asymptotes x = a, x = " a.

16. Find the volume generated by the revolution about either axis of the

hypocycloid x^ + y^ = d^.





CHAPTER V

RATIONAL FRACTIONS

33. A rational fraction is one in which the numerator anddenominator are rational integral functions of the variables.

The fraction is proper when the degree of the numerator is

lower than that of the denominator. If the degree of the

numerator is greater than that of the denominator, division can

be carried on until the remainder is of less degree than the

denominator. Suppose that Ny D are

rationalintegral functions

of X, and that the degree of N is greater than that of D, By

division,

D^^

D

in which B is of lower degree than D ; and, therefore,

7?In order to integrate the proper fraction " it is often neces-sary

to resolve it into partial fractions. It can be shown that

any proper rationalfraction can be decomposed into

partialfractions of the types

A B Cx-hG Ex + F

x " a {x"

aY 7?-\-px-{-q {oi?-\-px -{-q)*^

in which A, B, G, O, E, F are constants, r, s positive integers,

and x^ -\-px-{-q is an expression whose factors are imaginary.

For the proof of this and the related theory, reference may be

78



33-34.] RATIONAL FRACTIONS 79

made to works on algebra.* Here nothing more is done than

to work some examples in the principal cases that occur in

practice,t

34. Case I. When the denominator can be resolved into factors

of the firstdegree, all of which are reed and different.

7xg + 6a;-6

X

dx.x. 1. Find C^Jzl^jtllJ aj8 _ a;2 _ 6

Ondivision, x4--7x2 + 6a:-6^^^ 6(2x-l)3c8_aj2_6aj a8-a;a-6x'

6(20.-1).

"(x-3)("4-2)

Put6(2x-l)

Â^_

_C_(1)

x(x-3)(x + 2) X x-3 x + 2^^

in which ^, B, C are constants to be determined.

Clearing of fractions,

6(2x- l)=^(x-3)(x4-2) +Bx(x-h'2)-h Cx(x-3).

Since this is an identical equation, the coefficients of the same powers of

X in each member are equal. On equating the coefficients of like powers of

X, it is found that

A-{-B-{- C = Oj

-2l + 2P-3C=12,

- (M = - 6.

On solving these equations for A, B^ C, there results

^ = 1, J5 = 2, C=-3.

Therefore, after substituting these values in (1),

J x3-x*^-6x J\ X x-3 X4-2/

= ix2 + "4-logx + 21og(x-3)-31og(x

+ 2)

* See Chrystal's Algebra, Parti., Chap. VIII., Arts. 6-8.

t A few remarks on the decomposition of rational fractions are given in

Note A, Appendix.



80 INTEGRAL CALCULUS [Ch. V.

A shorter method for calculating A, B, C, could have been employed in

the example justsolved.

Q-

'

6(2g-l)Â

B C^^'^^^

x(x-3)(x + 2) x^x^s'^x^2

is an identity, it is true for any value of x.

Clearing of fractions^

6 (2x - 1) = ^ (X - 3)(x+ 2) +-B"

(X+ 2) + Oa;(x- 3).

On letting the factor x = 0,-4=1;

on letting the factor x " 3 = 0, orx = 3, B = 2;

and on letting the factor " +-2 = 0, or x = - 2, C = - 3.

Ex 2 r ^Êx.12, f '^^^

.^''- ^Jx-2-x-6 Jx2-4x+l

3 ri5x + l}^. Ex.13.f2a^-6x"-4x-ll^

Ex 4 fC4-3x)dx. Ex.14,fx^ + 7x" + 6x - 6^

^''* *"Jx2-3x4-2 J x2 + 2x

Ex 6 f-^. Ex.15. C--^"M^dx.Jx3-x Jx(x-i))(x + g)

Fx 6fM"l)^.

Ex.16,

f x"-3x + 3

^''- "- Jx(x2-1) .J(x-l)(x-2)

Ex 7 f Ca-5)x(toEx.17. fJ^+i)^.

E^ 8rC3x + l)dx

. Ex 18.f(8""-31x2+41x-6)dx,

^''- *"J2x2 + 3x-2 Jl2(x*-6x8+llxa-6x)

...,,|ai^. E.....jg::;f.:g[Suggestion. Assume the fraction equal to 1 + 1-etc.]



34-36.] RATIONAL FRACTIONS 81

35. Case II. When the denominator can be resolved into linear

factorj all of which are real and some of which are repealed.

Ex. 1.r6^-8x"-4x+l^

a;8-f a;2

in which A^ B, C, /", are constants to be determined.


6a;8 - 8a;2 _ 4 X 4- 1 =

-4x(a;-

1)2+ B(x-

1)2+ Gx^^x-

1)+ Dx^.

Equating coefficientsof like powers,

A + C = Q,

-A+ B-C+D=-S,

A-'2B =

-4,

B = l,

On solving these equations it isfound that J[ = " 2,-B

= 1, C = 8, i" = " 5.

Therefore,

r6^8_8x2-4x+l^^r/ 2 1_8

5_\^

Ja;2(a._l)a

^

J[ x^x^^x-^l (xîy)

=

-21ogx~i+81og(a;-l)

+ .

^

X x"1

Ex. 2. f-H^L-. Ex. 7. fiî^Zlii^..

")(x+ l)2 J (iC-3)2

Ex. 3. f(^-^)^. Ex. 8. f(^-1)^

.

J (X-3)2 "/X8 + 2X2 + X

2(fo;

a)8

Ex. 4. f2(x"21^. E^^ 3, r_a^J (2x+l)2 J(x +

Ex. 5. ff-" 5^-W. Ex.10, fiîi^cto.J Vx + a (X 4- 6)V J x8 - x2

Ex. 6. f 2^ Ex.11. f_i2x-_6}^_,

" (3V6 -2 -x)3 -^(aJ+ 3)cx + 1)"



82 INTEGRAL CALCULUS [Ch. V.

Ex.12, f ^^: Ex.14. f9(2 + 4x-x")da^,

Ex.13, f" ^^" Ex.15. fi?i"il^.J(x2-2)a J

x(x-l)8

36. Case III. When the denominator contains quadratic factors,

the linear factorsof which are imaginary. This case can be sub-ivided

into the two following :

(a)When all such quadratic factors are different.

(b)When one or more of them is repeated.

The latter case seldom occurs in practice. For each power,

from the firstto the nth, of these quadratic factors, a numerator

of the form Mx-h

N, in which 3f, N are constants, should be

assumed.

Ex. 1. Fmd f

^^^.

J x8 + 4 X

x(x2+4)'~x'

x2 + 4

Assume "

i=^ +

^"^-


4 = J[(x2+ 4)+x(5x+ C).

Equating coefficientsof like terms,

A + Bz=Oj

C = 0,

4^1 = 4.

The solution of these equations gives

^ = 1, B =

-l,(7 = 0.

Hence,

f_4^=ffl__ÛJx" + 4x JV" X2 4-4/

= logx-ilog(x2 + 4)

= log--^=.Vx2 + 4

Ex. 8. Find f ^"^^ " dx.J(x-2)(x2-2x + 3)2

Assume-^"1

"

=-^+.-g^+^

+"^* + ^

(x-2)(x2-2x + 3)2 x-2 x''^-2x + 3 (x2-2x + 3)2



35-36] RATIONAL FRACTIONS 83


"8+l= ^(x2 + 2x + 3)2+(5x4- a)(x-2)(a;2-2x+3)4-(I"x+Jg?)(x-2).

Equating coefficients of like powers of x,

A + B = 0,

-4^+0-45= 1,

10^ + 75-4C4-/" = 0,

-l2A-QB-^lC-{-E-2D= 0,

9A-QC-2E=l.

The solution of these equations is, A=l, B= " 1, 0=1, i" = 1, J" = 1.

Hence,

C (x^+l)dx^r/_i

^-1+ x"l \^

J (" - 2)(x2 2x + 3)2 J\x-2 x2 - 2x + 3 (x2 2x + 3)2;

= log ^1^2 ^ r 2dx

Vi2^2^M^2(x2 - 2 X + 3) J

(a;2_2 X +

3)2

The last integral can be found by means of reduction formulae to be given

in the next chapter.

2)dx

1)2Ex.3. fi^"J:}!l?I. Ex.6. (i^"^

J X8 + X J (x2-f

Ex.4. f 2X2 + X + 3^^ j,^ ^

r (2x2+ x + 2)dxJ (x+ l)(x2+ 1) J (3x + 2)(2x2 4-4x + 4)

Ex 5f3a^+6x8-fx2+2x+2^

^^ gr(3x2 "- 17x + 33)(fx

J 3x8+18x J x8-6x2 + llx

j,^ 3ry6-3x*-22x8+17x2-23x + 20

J (a;4.4)rx2 4-l)

dx.

Ex.10. (-^+7x^4-13^.J (X2+ 3)8

E^ 11r3x" + 3x^ + 30x2+17x + 75^

J Cx2 4-6)8

Ex.12, fî^c^x. Ex.13, f (^^-^)^Jx8-+-3x J x* 4- 6x2 4- 8

Ex. 14. ( -^^-^ dx = log-^"i-

+ i tan-i ?'.

Ja;8+ x24-4x + 4 (x + 1)^^

2

INTEGRAL CALC. " 7



CHAPTER VI

IRRATIONAL FUNCTIONS

37. The integrals of irrational functions can be found in only

a few cases. In some instances, by means of substitutions, these

functions can be changed into equivalent functions that either

are in the list of fundamental integrals, or are rational, and

therefore integrable. In other instances the integral can be

found by means of a formula of reduction. A few irrational

forms have been discussed inpreceding articles.

38. The reciprocal substitution. Sometimes the integration of

an irrational function is facilitated by the substitution

x = \dx=^-\dt,

Ex. 1. Find {y^^^dx.

On putting x =i, f ^^^îp^da;-- f(a^â !)*"""

(o""g 1)*

dx

Ex. 3

x-2Va2 - x^

dx

.2.

f ^ Ex.5, r

f-^. Ex.6, f^

Ex.4. |" ^- Ex.7, f^^^cto.84



37-40.] IRRATIONAL FUNCTIONS 85

Ex. 8. f ^ Ex.11, f ^

Ex. 9. f /^ .Ex. 12. f ^^^ " -.

"â;Va'^ - "" "^ (2ax- x^y

Ex. 10. ("^=' Ex. 18. f:^^"^cto."^ " Va;a - a^ J x^

39. Trigonometric substitations. Although the integration of

trigonometric functions is not discussed until the next chapter,

it may be stated here that trigonometric substitutions sometimes

aid the integration of irrational functions. The following sub-titutio

may be tried :

(a) x = a sin 6 for functions that involve-\/aF"3?y

(b)x=a tan 0 for functions that involve VoM^,

(c)x = a sec 6 for functions that involve Vo^^^^^^

Ex. Find (Va^x^dx. (See Ex. 1, Art. 28.)

On assaming x = asin^, (22;=:acos^d^,

and(Va^-x^dx =

a^(co"êde^f(1 + Co82 e)d$

2 2

40.Expressions

containingfractional

powers ofa

+6a?

only.If

71 is the least common multiple of the denominators of the powers

of a 4- hx, these expressions can be reduced to the form

F(XyVoT+bx),

If F(uyV) is a rational function of u, v, then jF(x, "y/a~+bx)dxcan be changed into a rational form by means of the substitution

a-\-bx

= ^.



86 INTEGRAL CALCULUS [Ch. VI.

For then x = ^"^, dx=:^^dz,0 0

and hence, Cf{x,VoT"S) d^ = îJV/'?lzi^,\^-'z.

Expressions that contain fractional powers of x only, belong to

this class. This is apparent on putting a = 0 in a-^- bx. If n is

the least common multiple of the denominators of the exponents

of X, the function can be changed into a rational form by the

substitution

x=z\

Ex. 1. Find f ^

"^ x^/a^ + bx

On making the substitution a^-\-

bx = z^, x =

^ ~ ^

,dx =

"dz.b b

Cdx

_ Q fdz

a z-\-

a

=

^log

^^'^-^bx-a

" Va2 + bx^-h a

.2. Find C~^^"d

"^1 + a* .

The L. C. M. of the denominators of the exponents is 6. If x = "*, then

dx=:6z^dz, and

= 6 x*(fX - io;*+ iX*- 1)+ 6 tan-i

x*.

Ex. 8. Find (-y^dx. Ex. 4. Findf 3 + 5(x^ + x*)dx^

-^* - 1

-^ 15(x + X*)




f"i+V^

,Ex. 9. f__"

__i5 ;

Ex. 6./"(a;i-2x*)(te Ex.10. C2ÊI1"1

dx.

j,^^ ^^V^-7-^+12v^^^ Ex. 11. J(3-x)V(F^r^

x(V^~\/x)

/^,

Ex.12, f ^^^.

Ex.8. lxVa+6x(Zx.'^(2x + 3)*

41. Functions of the form /{a?*,(a + fta;^)} . a?cfa;, in which w,

n are integers. If f(u,v)is a rational function of u, v, these

functions can be rationalized by means of the substitution

a 4- 6a^ = "**.

For thea, 2bxdx=nz''~^dz, x^= "

^^^,and the function becomes

0

Ex. 1. Find f ^^(See Ex. 10, Art. 38.)

This belongs to the form above, since"

X Vx2 - a^ x2 Vx2 - a2

On putting x^ " a'^ = z'^, xdx = z dz^ x'^ = z^-\-a\ and

Jdx _

C_dz_

=lton-iia a

a a a X

Ex. 2. f-^-^"- (SeeEx. 8, Art. 38.) Ex. 8. f ^^^.




Ex.4, r ^^ Ex.6, f ^^

Ex. 6. If /(", v)is a rational function of u, v, show that

4- ^'fâ'^can be rationalized by means of the substitution

^^"^ = """.

ex +d"

42. Functions of the form Fix, y^x^ + aa? + b)dx9 Fiu, v)

being a rational function of u, v. If the radical be Vma^+|M5+g,

it can be written Vm \a^-{-"x-^-"'

m m

The given function can be rationalized by assuming that

Va?^4- aa;-f

6 = " " a?,

and then changing the variable from a; to a;.

For, squaring and solving for ",

a + 22;

whence, " " a; = " -f "gH- o

and ,^ = 2^!"_^"^d..(a+ 20)*

Therefore, on substitution,

Ex." ^^

x\/a;2+ aj + 1

Assume vj^H-lc + l = " " as.

Squaring and solving for x, x =-=

i-.

1 4" 2 ^



41-43.] IHBATIONAL FUNCTIONS 89

Hence, ^^2(.^+ . + l)

(1+22)2'

and Vga + g; + 1 = g - x =

^^ + ^ + ^"

1 + 2"

On substitution,f

"

^= 2 f

-^^"^ " Vx2 + X + 1 ^ "2 _ 1

^.2?+l

_ iQg g-l+Va;2 + y + l

" + 1+ Vx2 + X + 1

43. Functions of the form /(", V-x^

"\-ax-\-6) da?, /(w, v)

being a rational function of w, v. If the radical be V"-mar^4-i"aj-f,

it may be written Vm\/-a^" " + -

^^ *^" factors of

" a? '\-aX'\-bare imaginary, V" aj2 4- aaj + ftis imaginary. For,

if one of the factors isx"a-^-i^,he other must be

"("""""/?),

and hence,

" a^H-aaj-|-6 = " (" " " + */?)("a "

t/?)

which is negative, and has an imaginary square root whatever x

may be. Only cases in which the factors of " a?^-f-

aa; 4- 6 are

real will be consideredhere.

Let"

0^4- "" + "=(""

a)(fi"

x).

Assume "\/"3i^-\-ax-fb

or V(a? a)(fi"

x)= (x "

a)z.

Then, squaring, /?" ar = (a? a)^*,




and V-a^ + aa;4-" = (a? a)2; = %^^ "

1 +2r

Hence, on making the substitutions,

which is rational, and accordingly integrable.

Equally well, the substitution, V(aj"

a)(/? x)= (fi x)z,

might have been made.

It follows from this article and the preceding article that, if X

is the indicated square root of an expression of the second degree

in X, every rational function / (a?,) is integrable.

Ex. 1. Find (- ^

Assume

V- x2 + 5 X - 6 = V(x - 2)(3 _

x) = (x - 2)2.

From this, on squaring, 3 " x = (x " 2)z'^,

Hence, ^ = '-^^

dx =^^^^,

and V-x2 + 6x-6=(x-2)g="^

"

'

Therefore, on substitution,

r dx_

_ 2C dz

Ja^V-x2+ 6x-6

J222 + 3

=

-V|tan-iV|"'

^3 "3 (a; 2)

Ex. 8. f ^ Ex. 8. f- *"

"' (1 + x2)\/n^ "' V2x2+3a; + 4




dx

Ex.4. (y!A^^"J^dx. Ex.6, r

Ex.6, f^ + ^

(to. Ex.7, r^^

xWl- x^

V6x-x^dx.

44. Particular functions involving "y/aix'^ftoc+ c.

(a)If a is positive,

r ^^

=-l=log(2aa;" + 2Va Vaaj^

-j-

fta?+

c)JVaa^ + 6x.+ c Va(Ex.43, Chap. III.)

(")If a is negative, say " aj,

r ^-^=JL

sin-i

^"i"^-^_.(Ex.44, Chap. III.)

(c)r (-4^tj)_^..

"/ V aar^ H- 6aj-f

c

Since " (cux^-{-

bx-{-c)

= 2 ax-\-

b,

dx

and Ax-\-B=:^(2aX'^b)-^B-4^,a 2a

"/ Vaar^ 4- fto?-f

c^ ^*^ Va^

"i-bX'{-c

\ 2a)J -y

dx

A

= " Vox"^-h

6a;+ c-h

an integral of the

a

form (a)or (6)above.

_, , ,,. ^ r (x + 3)(toEx. 1. Find i "

^^ ^^ "

J\/x2 + 2a; + 3

Since " (x2-j.2x + 3) =2a; + 2, and x + 3 = i(2x + 2) +2,

dx

C (x4-3)fte

_l.r

(2x + 2)dx C dx_

^ Vx2 + 2x + 3

~

2 J Vx-^+ 2x + 3 ^ V{x + 1)2+ 2

= Vx-^+ 2x + 3 + 21og(x+ 1 +"/x2 + 2x + 3).



92 INTEGRAL CALCULUS [Ch.VI.

Ex.3.)dx

"^ V5xa~2a;H-7 -^ V6-3x-2x2

Ex. 8. r^^

"

Ex. 7. (J'^"^dx,

Ex.4, f Êx.8. fi^xjLli^,

Ex.6, f ^Êx.9, f 7^ + ^

dx.

^V-2x2 + 3x + 4 -^ V3x^-3x+l

(d) f^ r

" ^"^="te,^ (x" a)

Voa^ + 6j;4- c^ {x "

a)Va^^f6"^fc

JIf,-^

being constants.

On putting a? " a =-,

da? = "

-^dz,andthe firstintegral takes

the form

-/;dz

in which A, B, (7, are constants. The first integral is thus

reducible to (a)or (h).

Since Mx-f--ZV=

3f(a; a)+N+ Ma,

.

r {Mx^N) ^^ rM{xâ) + N-^Ma ^^

"^ (x" ")Vaic^4-bx + c

^ (x "

a)Vao^ -\-bx-\-c

= mC^

4-(N4-Ma)C^

"^ Vax^ 4- 6a;-h

c "^ (a? a)VaJ^ + 6a?4- c

The two integrals in the second member have been considered

above.

Ex. 10. Find^ ^*

l)Vx^^"=^2x-f3

On putting x " 1 =-,

dx = "

-dz,and

z g2

f ^y-

=-f^^

=:-.J--log(gv^+v/rf2i^)"^ (x-l)Vx2-2x + 3

"^Vl + 2^2 V2

=Jl log

Va^'-2x + 3+V^_

V2 "-!




Ex. 11. Find ( {^^-^^)dx

"^ (aj- 2)V2x2 + 8x + 10

Since i^-"l=3 i-, and 2x2+ 8x + 10 = 2{(x + 2)2+ 1},

dx

(x+2)V2x2+8x + 10 V2'^"/(x+2)2+l -^ (x+2) V(x+2)'H1

The integrals in the second member belong to forms already considered.

Integration gives the result,

Alog(x + 2+Vx2 + 4x + 6)+2v^log^^' + ^^ + ^"i.V2 aj + 2

Ex.12,f (x-+ x + l)dx (suggestion:"^=x+4+.^.\"^ (x--3)"/5x'*-2(5x+34 \ x-S x-S J

Ex.18, f ^

"^ xVx^ + X + 1

Ex.14, f (2xi^l^^^.'^(x-l)Vl + 2x-x2

45. Integration of x'^(a-\-bx^)Pdx: (a) by the method of

undetermined coefficients; (b) by means of reduction formulae.

Some integrable functions are of the typex'^(a bafy,in which

a,b, m, n,

jp,are

constants.The

exponentn can

always be

positive. If the integration of an expression having this form

is possible, it can always be effected by the method of "inte-ration

by parts." A shorter method, however, may sometimes

be employed. The given integral is expressed in terms of a

function of x not affected by a sign of integration, and of

another

integral

which

is easier to integrate than theoriginal

function.

(a)EuLE. Put IQif^(aboifydxequal

to a constant times

one of the integrals

Car-''(a- bof^yx, jV+"(a-|-bx^ydXy

Car(a-\'^baf)^^dx,

Car(a+ baf^y^x.




plus a constant times x^+^(af "a?")'*+^,n which A, fi are the lowest

indices of x and of (a4- bx^)in the two expressions under the inte-ration

signs. Then determine the values of the constant coeffi-ients

thus introduced.

For example, on taking the firstof the four integrals referred

to in the rule,

Cx^(a-\-bafydx=Aar-''+\a-\-bo^y-^î-B Car-â-^-baf^ydx.(1)

Here A, ft, the lowest indices of x, (a-f 6af),under the sign

I,are m " n, p-, and from this the term

.4ar~"+^(a|-bx^y'^ s

derived. In order to determine the coefficients A, B, differentiate

both members of equation (1). The result, after simplifying, is

x"" =Ab (m -\-np -\-l)x -{ Aa(m

" n-f 1)-f

5.

From this, on equating the coefficientspf like powers of x,

j^^1

^^ a(m-n 4-1)

6(m 4- np -h 1)'" (m 4- np 4- 1)'

The substitution of these values in (1)gives

Similarly, by connecting iirf"(a- bif^y

x with the other inte-rals

mentioned in the rule, the following results are obtained :



45.] IRRATIONAL FUNCTIONS 95

/

+

an(p + l) J"''"("""'")^^^"fa^ [D]

In each of the four integrals with which jic"(a- bafydx may

be connected by the rule, m is either increased or diminished by

w, or else p is either increased or diminished by unity. The values

of m and p will indicate theone

of thefour

which is simpler than

Iaf"(a-j-hx"")"Xj and may preferably be connected with it. Suc-essive

applications of the rule are necessary in some cases.

(b)The results A, B, C, D, can be used as formulae of reduc-ion.

It is necessary only to make carefully the proper substitu-ions

for m, w,

p,

in them. It is notnecessary

to

memorizethese

formulae, as they can be kept for reference.

It is well to be familiar with the process of deriving A, B,

C, D, so that they can be readily obtained when required*

if necessary. Formulae A, C fail when m-h w/? + 1 = 0, B fails

when 771-f

1 = 0, and D fails when p -f-1 = 0. In these cases

other methods can be used.

Ex. 1. Find f ^^^.

Here, w = 4, n = 2, p = " J, a = a:^, 6 = " 1.

On connecting this integral with I "

^:^^^^^by the rule in (a),it follows

." -^ a//z2 x^that

* Another method of deriving these formulae, that of integration by parts,

is given in Note B, Appendix. Other formulae of reduction can be obtained

by connecting

\ x'^ (a-{"bx")Pdx vf'ithfx^-^-^Ca6a;'")''+*dxnd

(x'^+''-^(ahr')P-^dx

in the manner described in therule

in

(a).The

student may

derive the

formulae in this case as an exercis'fe. (See Edwards' Integral Calculus,

Art. 82.)




(1) C.^^=AxW"^^^-i-B^C^^^.

" thus,

(2) r x^dx^ îx Va^TT^ + Ci f

"/a2- a;2

^ Va^ - x*

Hence, on substitution in (1),

(3)C x*dx

=A^y/ai_^2 + BxVa^^^^ + C (^ -

"^ Va2 _ a;^ -^ Va-"-

x*

On differentiating and simplifying,

X* = 3^x2(a2 -

x2)-Jx* + B(aa- x2)-

Bx^ + C.

On equating coefficients of like powers and solving for A, B, 0, it is found

that

A =

-i,B =

-la^C=ia*.

Substitution of these values in (2) gives, since f = sin-i -"

f" ^^=r = - 13 a* sin-i?

-

x(2 x2 + 3a2)

Va23^2 \.

The coefficients A, Bi, might have been determined in (1),and î, Ci,

might have been determined in (2).

The integral could also have been found by the application of formula [A]

twice in succession.

Ex. 2. Find T"

^

Here m = 0, n = 2, p = " k, a = a^, 6 = 1.

Connecting the integral withf (x*-f a^)^^+^dx,

(1) f(x2+ rt2)kax = Axix^ + a2)-"+i+ b((x^+ a2)

"+idx.

On differentiating and dividing the resulting equation through by

(x2-a2)-*,l = ^(x2 + a2) + 2 Ax^(l - ifc) B(x2 + a2).

On equating coefficients of like powers, and solving for A, B, it is found

that

2o2(*-l) 2a^{k-l)




Substitution of these values in (1) gives

J (xa+ a2)*

"^

2 a2 (ifc 1)I(x^+ a2)*-i"^ ^^

* ~

^^J (x^+ a2)*-i*{This result might have been obtained by the application of formula [D].

Ex. 8. (Va^-x^dx.(SeeEx. 1, Art. 23, Ex. Art. 39.)

Ex.4.C ^^

.

lV2ax-x^ = x^(2a-x)^.]^ V2 ax - xa

Ex.6. f_*!^_. Ex.6, f ^.

Ex.7, f /^ .

"^ Va2 - xa -^ (^2 _

a.2)f'^ x8 Va^ - x^

Ex. 8. fX V2 ax - x^dx, [Suggestion. Put fx^(2a -

x)^dx

= Ax\2-

x)i + J?x*(2 -

x)* + Cx*(2a -

x)* + D(x~i(2a -

x)"*(te.]

Ex.9, f ^.

Ex.10. C ^'^^.

"^x*Vaa-xa "^(xa+ a^)*

Ex. 11. Show that

Jx"*(ixx'*-^ yga -- x^

. (w - l)aaC x^'dx

Va2 - x2"" t" J

Va^-aja

Ex. 12. Show that

fX- V^^:i?(to =^^^^^^^"^^^

+-ili~

f ^^^.

J w + 2 w + 2 J

-v/^nr^

Ex. 18. Show that

dx Va^ - x2, n-2 r dxdx

____

Vqa - x^ n-2 T

" x"A//i2_a-2(n - l)a2x"-i (n - l)a^J lx"Va2_xa (" - l)a2x"-i (n - l)o2j a?"-2 Va^ - x^

Ex. 14. Show that

r x"*dx__

x""-W2ax-xa (2m-l)qr x"* 'dx

"^V2 ox - x2*" m J

V2ax-x^



98 IN TEG HAL CALCULUS [Ch. VI.

MISCELLANEOUS EXAMPLES.

-Vxdx 15. f ^

r dx 16.r (a;-4)dx

8f____J^^?____. 17. f ^^

, ,/c",.o^+" 1 "ito partial frac-\

4rv/i^ + Vx-6^ (SeparateJ"

^^^j

"^ (X "

a)Vx " 6

"^V2x6 + 3x8 + 1

e. I '^^

xdx

'(x2-2)vGcir33

)dx

V-3x2-6x-2

dx

\/-27+ 10x+5x2

8.f V-^^ + ^^^-dx.

"^ (mx "

n)Vwix + n

22. f ^

9rV2x-x2(V2x-xHflr) + /t^^

'

J 34x-17x2 28. r

^^r(x''^ 3x+5)dx^

J(X + l)Vx2+l

20r (x- + 2x + 3)dx

" (x- + 2 X -" 3)Vl - a;-2

21. JV6H^"2(VSH"2x2+ax)2dx.

,2Va2-x2

x^dx

Va2 _ x3

10.fV2x-3(V2-3x+V2x+3)^^

"^ (2x-3)\/6-6x-6x224. J

11.finlîli)^

^^r

âx

"^ Vx2 + 2x + 426. J-p==:

Va2 - x2

12.r(^ + l)V^-2dx.

26. rxVa2

" x2

13. J(x-l)^(xiridx. 27. J^^'-^'cZx.

14. f_ (fX

i)\/x2TT

28. j'x2Va2-x2(ix.




J iK" va X ax. J ^^_ JJ2^^ _^

.

331 dx

86.* *^*

1)"

dx.80. f

'

'^ "

45CVl + x + x^

81. I.f

"

^ f V^"fa

47. f(a-''+6*-*'')V(a2-x")(a:'-6"

f- ^

(""" o")* 48. ^^V2ax-3fiers-" ?

(te.

84. j-(x""a")tcte.^^

r-x" + :^ +.3x

+ 9.

*

Jx" + 9x*+27x2 + 27'

85. f g"^

60. i "'-^^ dx.r xg + 6

^ Vx2+

!

\Vx2+2 VxM^ /

61. f ^^ + 2x+l^,

88. f ^^ "^V4x2 + 4x + 3

"

Jv2^^3:i^

^^r

x^+x-fj^^^89. Jv^^^^T^dx. "^Vx2 + 2x + 3

do f_^_ *^- f.

^'-^-3dx.

*^- J(x2+ aa)s-

"'V-3 + 12X-9X2

41f ^^ 64. r Vlx^ + mx + n dx, I negative.J (X*+ a*)2' -^

4Sf a;2dx

66. fJ(x8-a")2* -^(xa+a-

dx

(x8 a")2*-^ ("*+ o^)

Vxa - a^

(x+l)dx43. f ^

66. f-J (x2 2 X + 3)2 -^ (x2-2x + 3)2 -^ (2x2-2x+J)V3x"-2x+l

INTEGRAL CALC. " 8



CHAPTER VII

INTEGRATION OF TRIGONOMETRIC AND EXPONEN-IAL

FUNCTIONS

Integration by parts and the use of substitutions will be found

very helpful in obtaining the integrals of trigonometric and

exponential functions.

46. Isin* xdai"9 I eo8**a?"la?9 n being an integer.

(a)If w be a positive odd integer, say 2m + 1,

Isin"a;cto= jsin**+â;daj= jsin** a? sinxdx

= " I(1"

cos'o;)"'(cosx).

The binomial in the latter form can be expanded, and the inte-ral

found term by term.

Ex. 1. Isin*xdx = " f(l--os* x)d(co8x) = - cos aj + J cos* " + c.

Similarly,

jcoa^'^+^xdx= I(1" sin*")""(:?sina?) sin a?"

^ sin*a? +""".

Ex. 3. (cos^xdx=

({l-am^xyd(8inx)= ((1-2 Bm^x-\'8m*x)d(Bmx)

= sin 05 " }sin"x+ 1 sin* a; + c.

(b)If n be any positive integer, integrate sinâcZa? by parts,

putting

w = sin'*~âj,d'y= sinajdaj.

100



46.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 101

Then, du= (n " 1)sin**^ x cos xdx, v = " cos x ;

and Isin'^xdx=

"

sin**"âjcos a? + (n"

1)jsin"~â?cos*ajda?

= " sin*-â?cos x+(n" l)(sin^-â?(1 sin*a?)daj

= " sin**~â;cosa?-|-n " 1)|sin*"â?dic

" (n"1) Isinâjda?.

By transposing the last term to the first member, combining,

and dividing through the equation by w, the result is

fsin'xdx = - si^-'^cosx^^L^Jsi^n-.^^.A]

The result A can be used as a reditction formula. Successive

applications of it leads to- I da? or I sin a;da? according as w is

even or odd.

Ex. 3. jsm^xdx=-.^^^^^^ysmxdx

1 2= " - sinô; cos x " cosas + c.

This result may be compared with that of Ex. 1.

On integrating cos"a?c?a? by parts, putting u = cos**"â?, dv =

Gosxdx, the following reduction formula is obtained,

/'^

, sina;cos'*"â? n " l r^

" _ r-Ri

cos" xdx = "

I cos**~^ x dx. Ii^Jn n J

The deduction of B is left as an exercise.

(c)Suppose that n is a negative integer.

The value of jsin"^

a; (fa?in A is

J n " 1 n " lJ



102 INTEGRAL CALCULUS [Ch.VII. 46-

On changing n into n-|-

2 this becomes

/.-,

, sin""*"*cos a?,

n

-f

2 /* .

". 2 ^ rî

sin" xdx=^

'-"

I sin'*^* x dx, fC |

This can be used as a formula of reduction when n is a nega-ive

integer.

Ex. 4. f^^

= f(sinx)-8 da; = - i cos x(8in")"* + f(sinx)-i dx

cos a;,

r^""^j^= 7"

"

h I CSC X dx

1 X

= " - cot as CSC as + log tan - + c.

2 2

Similarly, on solving for Jcos^'âjcfa?in B, and then changing

n into n-H

2, there results,

fcos'xdx-'J^^^^^

n"4fcos-^^xdx.D]J n + 1 n + lJ'--'

This is a formula of reduction for jcos"a?(ir when w is negative.

It is advisable to remember the method of deriving A, B, C, D,

so that they may be readily obtained when necessary.

Ex. 6. (a) icos^xdx, (6) rsm*a;dx,(c) ism^xdx,

Ex. 6. (a) rsin*a;da5,(6) isin^xdXj (c) Tsinâjcte.

Ex. 7. (a) (cos*; da;, (6) Icos'âjda;, (c) (cos^xdx.

(*)JJ-.'^"Jife'^^L-fe-

Ex. 9. Show that f^sin2""(to^ '^'^""^^^~ ^^

.^.Jo 2.4.6...2W 2

IT

Ex, 10. Showthatpsing""+â;(fa;=^'^'^-^m

.

dx

cos^x




47. Algebraic transformations. The trigonometric integral

Isinâjcôjan be put into an algebraic form. For, if

sin x = z,

then cos xdx = dz,

and da: =

-^^-

^^-

cos a? VI " 2*

and hence, (sin"ajcto = (^ ^

"

The second member has a form which has been discussed in

Chapter VI.

If the substitution cos a? = 2

be made, |sin" xdx = " (1"

"*)* da;.

This form can be integrated by methods already explained.

These substitutions may also be employed in the case of 1 cos" x dx,

Ex. 1. rsin*a;cfcc.(Compare with Ex. 3, Art. 46.)

On putting

" = sinx, isin"a;(a;= I - = Vl - "2 + c

= " J sin* X cos " " } cos x + c.

Ex. 2. Solve Exs. 2, 4, 6

(a),7

(a),8

(6)ofArt. 46 by

algebraic sub-titution

4a Jsec" a? dac, Jcosec"" x dx.

(a) Since sec a? =, and cosec x =

-:"

,I sec" x dx and

/cos

a; sinx J

cosec" a? da; can be reduced to the forms considered in Arts.

46, 47.



104 INTEGRAL CALCULUS [Ch. VII.

Ex.1, (a) fsec*r (te, (6) fcosec^xdx,c) f-^, (d) f" ^"

[SeeExs, 8 (c),8 (^r), (d),6 (a),(Art.46).]

Ex. 8. Find f^

^

, ^"assuming x = atan ^. (See Art. 39, and Ex. 2,

Art. 46.)

(b) If n is an even positive integer, another method may be

employed.

Since sec* a? = 1 + tan* x, and d (tanx)

= sec' x dx,

|sec"ajdaj=sec"~*a;sec*a;daj

=r(l+ tan"aj)^~dtanx).

The binomial under the sign of integration can be expanded inn " 2

a finite number of terms since n is even, and accordingly"

^"

is an integer.

In a similar way it can be shown that

Icosec"ajcfaj = " |(14-cot*aj)d(cotx).

Ex. 3. f8ec" X dx = r(1 + tan^ x)2(i(tan);

= tan X + } tan* x + J tan^ x-f

*

c.

(Compare Ex. 8 ("),Art. 46.^

Ex. 4.' (a) isec*xdx; (6) lcosec*X(ix;

(c) i cosec" X dx ; ("i)i sec*?

(?x.

[Compare the results with those of Exs. 8 (c),8 (/), 8 (g).Art. 46.]

(c)If n is any positive integer greater than 2, the method of

integration by parts can be used.

On putting sec"

*

a? = w, sec* xdx = dv,

it follows that du = (n " 2)sec*** a? tan xdx^ v = tan x.




Hence, jsec" xdx= jsec**"^ x sec^ x dx

= tan X sec"'^ a? " (n" 2)jsec**~^ x tan^ a? da?.

On substituting sec* a? " 1 for tan* a? in the last term, and solv-ng

for Isec** X dx, there is obtained

fsec-;tdx*5EL^^5^ fsec-'a^da..A]J n " 1 n " lJ

This formula of reduction leads to |sec a? cto?when n is odd,

and to Idx when n is even.

Similarly, integration by parts will give

/-, cota?cosec'*~*aj ,

n " 2 /*"

-j rT"n

cosec" xdx = 1 I cosec"*x dx, [B]

n " l n " 1*/

which, on repeated applications, leads to |cosecajda? or to I dx,

according as /i is odd or even.

Ex. 6. (a) Tsecxdx; (b) isec^ xdx; (c) I cosec^ x dx ;

(d) leasee^ xdx; (e) isec^^xdx,

[Compare the results with those of Exs. 8 (6),8 (d),4, 8 (a),Art. 46.]

Ex. 6. (a) laec^xdx; (6) fcosec*a;cto.[Compare Ex. 4(a), (6).]

(d)Transformation to an algebraic form.

dzIf tan a? = 2!, a? = tan"^2!, dx = -

-, sec*

and hence, I sec" a? da; = j(1+ 2*)*

(fe.

Also, if sec a; = z,




"

dzit follows that dx =

and hence, I sec* xdx=:( ^

dz.

In like manner, the substitutions z = cot x, z = cosec x, will re-duce

I cosec" a; c?a;to an algebraic form.

Ex. 7. Solve Exs. 3 (a),4 (a),4 (6),1 (d)above, by algebraic substitution.

49. Itan** x dx^ i cot^ x dx.

(a) Let n be a positive integer.

Then,Itan** a; cfo; |tan""* x tan* a? da?

= Itan*~* a? (sec*? " 1)cto

= Itan"*a? d (tan")

" jtan"~*a?daj

^ tan;^ ftan"-*; da;. [A]n " 1 J

This reduction formula leads to idxoY to jtanajc^, accord-ng

as n is even or odd.

N

Ex. 1. (tan*xdx=z

Ttanx (sec^

"

1)cZx=

TtanasJftanx)"

Ttanx dx

= J tan2 X " log sec x-}-

c.

In like manner,

Icot" a; da? = j cot"~* x cot* a? da; = I cot"~* x (cosec*; " 1)da;

= -

52^^fcot"-*dx, [B]

n " L J

another reduction formula.



60.] TRIGONOkETRlC AND EXPONENTIAL FUNCTIONS 107

(b) If n is a negative integer, say " m,

Itan"a?da?= " cotôjcto?,and lcoVxdx= (tan'^xdx.

Hence, this case reduces to the preceding.

Ex. 2. (a) itAii^xdx, (6) \ta,n^xdx, (c) itaji^xdx,(d) (cot^xdx,

(6) icot^xdx, (/) (coi^xdx:

(c)Transformation to an algebraic form.

dzIf tan a; = 2, dx =

:^""

-',

and hence, " tan**a?da? =

ij^'dz+ z'

Again, if sec a? = a;, da?^

z-yjz^1

'

n-2

and hence, I tan" xdx=^ i " dz.

Similarly, I cotâjcâ?can be changed into algebraic forms by the

substitutions,

z = cot X, and z = cosec a?.

Ex. 3. Solve Ex. 1, 2(a),2(d) above, by algebraic substitution.

50. jsin***a? cos" ficdJaJ.

(a) When either m or n is a positive odd integer, no matter

what the other may be, the form sin"*x cos" x dx can be integrated

very easily. For, it is then reducible to a sum of integrals of

the form jsin^ a? d (sina;),r of the form I cos* a? c?(cosa;).This

is illustrated by the following example.



108 INTEGRAL CALCULUS*

[Ch. Vn. 50-

Ex. 1. sin' X cos" xdx= TsinT (1 " sio*x) d (sinx)

=

^ sin

7x "

^ sin^ x+c.

Ex. 2. fsin*xcoe^xdx. Ex. 4. fJ^BiBLcfe.sin* cos* X dx. Ex. 4. i

/C08X

Ex. 8. fcos*X sin* x dx. Ex. 6. f?^^ "to.

"^ -^v'snri

(b) When m + n is a negative even integer, say ^2p,

sin* X cos" a; da? can be integrated by means of the substitution

tan x = L

If tana; = ^, da? = -"

r, sina? = "

.cosa; = " :

! + "" Vi+T* VTT?

and hence, | sin- a; cos** a? cto? ) ^^^^^-= jr (1+ ^'"^d^.

The last form is readily integrable. Sometimes the substitu-ion

cot x=t is better than the substitution tan a? = ^ for obtain-ng

a simple integrable form.

rsing X^

J C08*X

Iftanx = f, (^^dx=(îl + t^)dt ifi+ ifi+ e

J COfi^X ^

= J tan* X + i tan* x + c.

The actual substitution of the new variable may often be conveniently

omitted ; for instance,

f!i5l^"2x=ftan2xsec*x"te=tan^ x (1 -htan^ x)d(tanx)

J cosP X J J

= \ tan* x-\-\tan* x -f-c.

Ex. 6. fî^dx. Ex.7. f^(te.J cos'' X J sm" X

(c)Transformation to an algebraic form.

dzIf sinaj = 2;, cos a? = vT^-"^, dx"

^

and hence, J sin" x cos" a; da?=1 2"'(1 i^^dz.




In like manner the substitution z = cos x will give

J sinâjcosâdajrs J 2J*(1"2;^*

dz.

The substitution tan x = t leads to a simple form when m + n

is a negative even integer. This has been shown above. See

Ex. 6.

51. Integration of sin^ x cos^ x dac i (a) by the method of

undetermined coefficients; (b)by means of reduction formulae.

(a) Rule. Put jsin*"a? cos** a; da? equal to a constant times

one of the four integrals,

jsin"*~^X cos" X dx, isin*"x cos**"^ x dXy.

Isin""^^ cos** a?da?, |sin"*a? cos**"*"^ a?da?,

p?ws a constant times sin''^^ cos*"*"' a?, in which p, q are the lowest

indices of sin x and cos x in the two expressions under the inte-ration

sign. Then determine the values of the two constant

coefficients by differentiating, simplifying, and equating coeffi-ients

of like terms. For instance, using the first of the four

integrals referred to in the rule,

Isin"*X cos** xdx= A sin""^ x cos***"^ x + B I sin*"""^cos" xdx.

Here the lowest indices of sin x, cos x, under the sign I,

are

m"

2, n; and from them the term ^ sin"*"'a; cos""*"^ a; is formed

by the rule. On finding the differential coefficientsof both mem-bers

of this equation, there is obtained

sin**X cos** a; = (m " 1)^ sin*"^

x cos**"*"â; " (n-f-1) sin**x cos** a?

+ B sin"*~^X cos** x.

Divisionof

both terms bysin"*"^

a? cos** a;

gives

sin*a?= (m " 1)-4(1 sin*a?) (n-f-1)-4 sinâ?-f-

5.



110 INTEGRAL CALCULUS [Ch.VIL

On equating the coefficients of like terms in both members^

" (m-I-n)-4

= 1,

(m-l)^ +-B

= 0;

whence, -4=

,B = "

Hence,

J,^ ^ ,

8in*^-*a5COB**+*a5fn + n

+ ^^=if8ln"~-"? C08~ 05 da?. [A]

In a similar way, by connecting I sin*" cos* a? c2a?with the other

integrals mentioned in the rule, the following reduction formulae

are obtained:

ssln*^ X COB" xdx =sin*"

+* a? 008**+ *

05

m + 1

^w + n

-f2 rsinmâa.co8"a? da?, [B]

ysin"* 05 cos" xdx =

sin"*"*" 05 cos**~^ 05

+_WLZil.fsinmajcoS"-2o5claj.

[C]

J'in"* 05 cos" xdx = -

sln"*+ "

05 cos" +* 05

n + 1

+

^î^^fsin"*ajcos"+"arcla5.D]

n + 1 ^

In each of the four integrals with which I sinâjcosâjc^ may

be connected, by the rule, m or n is increased or diminished by 2.

The numerical values of m, n will indicate the one of the four

which is simpler than |sin**; cos" a: c?a;,and with which it may

preferably be connected. A succession of steps like A, B, C, D,

may be necessary.




In solving the problems of this and the following article,it

may happen that the results will not agree in form with those

given in the answers. An agreement can be made by using the

trigonometric relations, sin^ a;-f

cos^ a? = 1, secâ?= 1-h

tanâ:, etc.

Any apparent difference in results will be due to a difference in

the methods of working the examples.

Ex. 1. (sin^ z cos^ x dx.

Assume Jsiu^ a; cos^ xdx = Asmx cos^ x-\- Bi cos^ x dx.

Differentiating, sin^ x cos^ x = A cos* x "

îA sin^ x cos^ x-^ B cos^ a,

whence, dividing by cos^ ", sin^ x = A(l " sin^ x)" ZA sin^ x-^ B,

Equating coefficients of like terms,.

-4A= l,

A + B = 0.

On solving these equations, A = " ^, 5 = J.

Hence, jsin^ x cos^ xdx = " Isinx cos^ ^c + } i cos^ x dx ;

from this, by Art. 46, = " J sin;c cos^ " + j (sinx cosx + x)+ c.

Equally well, I sin* a; cos^ " dx might have been connected with Jsin^xt^x.

Also, 1 " cos^x might have been substituted for sin^x, or 1 " sin^x for cos^x,

and the integral found by the method of Art. 46.

Ex.2. rsin*xcos2x(te. Ex.4. C^^dx.J J sm* X

,.8. (*"-^"

" Ex.6. f5J sm* X cos* X ^8

Ex.3. 1 "^ Ex.6. \^^dx.' 'sms X

Ex. 6. Solve some of these examples by reducing them to an algebraic

form, as described in Art. 60 (c).

(b)The results A, B, C, D can be used as formulae of reduc-ion.

It is necessary only to substitute in them the proper values

for m and n. It is not necessary to memorize these formulae.

The student should make himself familiar with the process of

deriving these formulae, so that he can readily obtain them when




required* The formulae A, B, C, D of Art. 46 are special cases

of A, B, C, D above. This will be apparent on putting m and n in

turn equal to zero in the latter formulae. Moreover, jtan" a?da?,

jcot" a? da?, discussed in Art. 49, may be put in the forms

Isin" a? COS"" a? daj, I cos" a? sin"" a;da?,and solved by the methods

of this article. For the sake of practice in making the substitu-ions

a fewexamples may

besolved

by means

ofthe formulae.

Ex. 7. fsm"a;co8*"daj.

By^, fsinea;co8*ajda;-^^^^^?^^^

Af8m*xco8*x"to;J 10 lOJ

by^, f8in*xco8*a;da;-5yî^^2^

+ |f8m2a;cos*x(te;

by^, j'sm"aico8"a;(fa-^"''""^'' + lj'co8"x(te;

by a, j'co8"xdie^M^ + |j'oo8"*"fe;

byC, fco8"*(to= ?iM^ + lf"te=5E^"2S*i" + c

The c6mbination of the results gives

Tsin" cos* xdx = - ^ sin^ x cos*x

" ^ sin* x cos^ x - ^ sin x cos^ x

+yJj

sinx cos^x + ^ sinx cosx + ^x + c.

The formulsB might have been applied in other orders, for example CACAA,

CCAAA, etc.

Ex. 8. Solve Exs. 2, 3, 4, 5 by means of the reduction formulse.

* Another method of deriving these formulse, namely, by integrating by

parts, is given in Note C, Appendix. Other formulse of reduction can be

obtained by connecting

I sin* X cos" X dx with I sin'"-2x cos""+2 x dx, isin""+2 x cos"-* x dx

in the manner described above. The formulse for these cases may be derived

as an exercise. (See Edwards, Integral Calculus, Art. 83.)




52. Itan*^ no sec** x dx, |cot^ x cosec*^ x dx.

(a)Reduction to the form jsinâjcosôjda?. This may be done

by the substitutions

sin a? 1i

cos a; 1tajia;=

,seco; =

, cot x =-:

"

,cosec a; = "

cos a; coso? smo; smo;

and the integration can then be performed by one of the methods

of the last two articles.

(b)Reduction to an algebraic form.

If tan a; = 2?, I tanâ? sec^'xdx = j2;"*(1z^^ dz.

This is almost immediately integrable if n is a positive even

integer. In this case, I tan*" a? sec" a; (fa?can be reduced to inte-rals

of the form I tan''a:d(tAna?).SeeEx. 1, below.)

tan*" X sec" xdx= 1 2!"~^ (2^ 1)^ dz.

This is almost immediately integrable if m is a positive odd

integer. In this case, I tan*"a; sec" a?da? can be reduced to integrals

of the form |sec* a;c?(seca?).(SeeEx. 1, below.)

The form I cot'*ajcosec"aj dajmay be treated in a similar manner.

Ex. 1. Ttan^xec*xdx=^

Ttan* sec^x

seci^xdx

= (taiTi^x(tm^x+l)d(i9,nx)

= Jtan"x + Jtan*a; c.

Or, Itaii^ X sec* x dx = I tan^ x sec^ x sec x tan x dx

= i (sec2"

1)sec* x d (secx)

= i sec^ X " J 8ec*x + c.



114 INTEGRAL CALCULUS [Cii.VII. 62-

Ex,3. fsec"g(to, g^ g fcot6xco8ec"x(te-J lan8 X J

Ex.8, ftan^xsec^xda;. Ex. 6. ftaiiTx8ec"xdx.

Ex.4, i cot^ X cosec* X dx. Ex.7. I tan* x sec* x (to.

Ex. 8. Solve some of these examples by using algebraic transformations.

53. Use of multiple angles. When m and n are positive and

one of them is odd, the firstmethod of integration shown in Art.

50 can be employed in the ease of I sin* x cos" a da?. When m and

n are positive and both even, the use of multiple angles will

aid the process of integration. The trigonometric substitutions

that can be employed for this purpose are :

sin X cos a; = ^ sin 2 a;,

cos* a? = ^(1+ cos 2a?),

sin*a;= ^(1 " cos 2aj).

Ex. 1.fsin2

X cos^ x dx =

JJsin^2 x dx

= JJCl co84x)dx

= Jx " ^8in4x + c. (Compare Ex. 1, Art. 61.)

Ex. 2. fsin*x dx. (SeeEx. 6 (a),Art. 46.)

Ex. 8. fcos* X dx. (See Ex. 7 (a),Art 46.)

Ex. 4. fsinsdx. (SeeEx. 6 (6),Art. 46.)

Ex. 6. fcos"xdx. (SeeEx. 7(6),Art 46.)

Ex.6. fsin*xcos2xdx.(See Ex. 2, Ar*. 61.)

Ex. 7. Tsin* cos* X dx.




54f ^

.

On denoting the integral by /,and

dividing thenumerator and

denominator by cos^ x,

j__r sec'xdx

_

r d{t\""

J a^ 4- 6Han2 a;

~

J a^ 4- "

:an x)Wt^,u^x

On substituting u for tana;, integrating, and replacing u by

tan X, there is obtained,

ab \ a J

eer dx C doc

J a + 6 COS ac' J a + 6 sin a;*

Since cos x = cos^ ^" sin*^,nd cos^ ^+ sin^^= 1"

c ^=r

'.

^ m

On dividing numerator and denominator in the second member

by

cos*^nd reducing,

sec* ?dx

r djx^

1 r 2

Ja + "cosaja^bJ ^^^

.^ + ^

2 a-6

(fAan^^

^-"^"^tan^^ + ^Li:^2 a " 6

"1 oy

^^ 11"

T5" 3.ccord-

ing as a is greater than b, or less than b. Hence,

ifa"", f" ^=

_l_tan-Y\^tan2Va + 6 cos a;. Va* " 6* V a + ^

V

INTEGRAL CALC. " 9



116 INTEGRAL CALCULUS [Ch.Vn. 55-

ifa " 6, I "

-^

= "

,

log "^""-

=

_l_tanh-fJ|H"tan?YOn introducing the half-angle as before, and dividing numerator

and denominator by cos* % as in the case justconsidered,it

will

be found that,

J a-{-b sinx J

dx

a/'cos^lsin^l'j2 6 sin|cos|

=/:sec^^da;

a-f-

2 6 tan I-ha tan*?

a J I

This is in the form 1"-?".,or f^

^

^ according as a is greater

J z^-^-crJ z^ " cr

or less than b. Hence,

if a " 6, r"

"^.

" =^

tan-^

J a -h 6 sma;

-^^2_

52

a tan - + 5

a tan -

-f6 " V o'* " a^

if a " 6, I "

-^" :=

,

log

Ja-hbsmx V6*-a* % tan ?+ 6 + V6^^^

V6*-a*: coth-i

atan^+6




In working the examples it is preferable to follow the method

employed above, and not to use the results that have justbeen .

found as formulae for substitution.

Ex. 1 f "^*-

v^ A C dx

,.1.f ^^ . Ex.4, f"J3-2s!nx J6- 3cosa;

^"2. f^ f^. " Ex.6, f" -

J2 + 38in2x J 4 + 5 cos 2 X

8. (- p Ex.6, f"J5 + 3COSX J 4-+ 5 sin 2 "

Ex. 7. When a = 6 in this article,find the integrals.

56. \")^"innxdXf (eêosfixdx.

On integrating e'^ sin nxdx by parts, first taking e^dx for dv,

and then taking amnxdx for dv, thereare

obtained

Te"smnxdac =

^'^^^^- - fe osnxdx, (1)

Ce^smri^dx-^^-^^^^-h-C^cosnxdx,

(2)

The integral in the second members of (1),(2)can be elimi-ated

by multiplying the members of (1)by - and the members

n

^

of (2)by -, and adding the results. When this is done it will

be found that

/'"-sinr^d^ = '^^^(^^^^^^^^21^. (3)

Similarly, on integrating e"*" cos nxdx by parts, first taking

e'^dx for dv, and then taking cos na; da; for dv, and eliminating

I e*** sin nx dx which has thus been introduced, there will be

obtained

J a* + ""^ ^



118 INTEGRAL CALCULUS [Ch. VII.

The result (4)can be obtaiued by eliminating ief^^mnxdx

from

(1)and (2).It can be deduced

also

by

substitutingthe

result (3)in (1)or in (2). It is, however, preferable to deduce

it directly by integrating by parts.

As in Art. bb the student is advised to work the examples

by the method followed above, and not to use (3)and (4)as

formulae for substitution.

X. 1. fe'sinxdx. Ex. 4. J-x. 1. Je'sinxdx. Ex.4.

\^2^dx.

Ex. 2. Je'cosxdx. Ex. 6. f^"to.

Ex. 3. (e^cm^xdx. Ex. ". ie'Go^^xdx,

57. Tsinrnxos n^vcf a?, rco8fna?co8na?c7;v,Jsinm^vsiima^c

Since, by trigonometry,

sin mx cos nx = ^ sin (m-{-n)x+ \ sin (m "

n)x,

/,cos (m

-{-n)xcos (m "

n)xsin mx cos nxdx = "

"

-)!"

-^^ -^4"

2(m +

n)2(m "

n)In a similar way it can be shown that,

/,sin (m

-{-n)x sin (m "

n)xcos mx cos nx dx =

"-i" "

-^ -\"

-i^,

2(m + 71)^

2(m-w)

/,

sin (m 4- n) x sin (m "

n)xsm ma? sin nx dx =

^r^^" " " f"

-\^ ^

2(m4-w)

^

2(m-n)

Ex. 1. jcos 3 a; sin 6 X dx. Ex. 4. icos 3 x cos Ja;dx.

Ex. 2. rcos4xcos7xdx. Ex. 6. fcos |x sin Jx dx.

Ex.3. (sin5xsin6xdx. Ex.6, fsiny'^

x sin^s^y

dx.



CHAPTER VIII

SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS

58. Successive integration. It has been seen in the differential

calculus that successive differentiationith respect to x is some-times

required in the case of functions of the form

%=/(a?);

and that successive differentiation with respect to both x and y

may be required in the case of functions of the form

u=f(x,y).

On the other hand, the reverse process called successive inte-ration

is sometimes necessary. This chapter will be concerned

with describing the notation that is used in "multiple integrar

tion,"as it is

often termed ; andit

will show,by

examples,how

successive integration is introduced and conducted. Arts. 61,

62, 63, contain applications of multiple integration to the measure-ment

of areas in rectangular coordinates, and of volumes in rec-tangula

and polar coordinates. Plane areas in polar coordinates

and ciirvilinear surfaces will be found by means of multiple

integration in Arts. 67, 75.

59. Successive integration with respect to a single independent

variable.

Suppose that fii^)= |/(^)^^" (1)

f2(x)=fA(x)d^, (2)

119



120 INTEGRAL CALCULUS [Ch. Vlll.

and M^)=fMx)dx. (3)

Since /j(x)

=f[fi")]x,

itfollows from (1)that f^(x)=C [Cf(x)dx]x ; (4)

and since f^(x) j\f^(x)\dx,

it follows from (4)that /g(a;) fjf[Cf(x)dx]x ]-x, (5)

The second member of (4)is usually written in a contracted

form, namely,

jjf{x)dxdx,r

^jf{x)^, (6)

in which da? means {dx)^,nd not d (a?).

Similarly, the second member of (5)is usually written

CCCf(x)dxda:dx,r CfCf(x)da?, (7)

Integral (6)is called a double integral, and integral (7)is called

a triple integral. In general, if an integral is evaluated by means

of two or more successive integrations, it is called a multiple in-egral.

If limits are assigned for each successive integration, the

integral is definite ; if limits are not assigned, it is indefinite.

Ex. 1. Determine the curve for every point of which the second differ-ntial

coeflBcient of the ordinate with respect to the abscissa is 8.

The given condition is expressed by the equation

(1)^

= 8.

"mdx

whence, d(^] =

\dorJ

This may be written-^

" = 8 ;

Sdx.



59.] SUCCESSIVE INTEGRATION 121

Integrating, (2) ^ = 8 x + c,

dx

whence, dy = (Sx -^ c)dx.

Integrating again, (3) y = 4 x^ + ex + k.

This is the equation of any parabola that has its axis parallel to the y-axis

and, drawn upwards, and itslatus-rectum equal to 4. All such parabolas will

be obtained by giving all possible values to c and Jfc,he arbitrary constants of

integration. Two further conditions will serve to make c and k definite.

For instance, suppose that the tangent to the parabola at the point whose

abscissa is 2, isparallel to the x-axis ; and also that the parabola passes through

the point (3,6). By the former condition,

^= 0whenx = 2;

dx

and hence, by (2), 0 = 8 . 2 + c,

that is, c = " 16.

Equation (3)then becomes y = 4 x^ " 16 x + A.

Also, since the parabola passes through the point (3,5),

6 = 4. 32-16. 3 + A;;

whence. A;= 17.

Therefore the equation of the particular parabola that satisfiesthe three

conditions above is

y = 4x2-16x + 17.

The given relation (1)might have been written in the differential form,

d^ = Sdx^,

and y expressed in the form of a multiple integral, namely,

whence on integrating, = ( (Sx + c)dx

= 4x2 + cx + A;.

The former solution is better because it shows all the steps more clearly.

Ex. 2. If 8 represents distance measured along a straight line, and t time,

" is the velocity of a body that moves in the straight line, and " is its

dt dfi

acceleration or rate of change of velocity. In the case of a body falling in

a vacuum in the neighborhood of the earth^s surface, the acceleration or rate



122 INTEGRAL CALCULUS [Cii.VIII.

of increase in the velocity is constant and equal to about 32.2 feet-per-second

per second. The number 32.2 in this connection is denoted by the symbol

g. Let it be required to determine 8 from the known relation,

'(I)his may be written " =- " = g.

Using the differential form, "i(-)= fl'*"

da

and integrating, (2) " = gt + c,

at

in which c is an arbitrary constant of integration.

Writing the latter equation in the differential form,

d8=("gt^c)dt,

and integrating, (3) " = ^ gt^ + c" + Jfc,

in which k is another arbitrary constant of integration. In order that the

constants c, k may have definite values, two further conditions are required.

For instance :^

(a) Suppose that the body fallsfrom rest, and that the distance ismeasured

from thestarting point.

In this case, s = 0, and " = 0, when * = 0.

Hence, substituting in (2), 0 = 0 + c,

that is, c = 0 ;

and, substituting in (3), 0 = 0 + 0 + ik,

that is, ik= 0.

Therefore, the distance through which a body fallsin a vacuum on starting

from rest is \ gt^^ in which g is about 32.2 and t is the duration of fall in

seconds.

(6) Suppose that the body has an initial velocity of 8 feet per second, and

that the distance is measured from a point 12 feet above the starting point.

By the last condition, " = 12 when t = 0;

and hence, by (3), 12 = 0 + 0 + jfc,

whence k = 12.



69-60.] SUCCESSIVE INTEGRATION 123

By the other condition, " = 8 when " = 0 ;" dt

and hence, by (2), 8 = 0 + c, that is, c = 8.

Therefore, under these conditions,

8 = \gV^-\.^tJr12.

The known relation (1)might have been written in the differential form,

d^s = gdt^ ;

from this, * = i\9dt^

whence, on integration, " = \{gt-\-c)dt

= hgt^+ ct + k.

Ex. 3. Evaluate C C C^ (dxy.

The integrations are made in order from right to left. Thus, if / denote

the integral,

'=n"[?j:(-'-rf"*"'

= 16.

Ex. 4. Evaluate CCCoi*(dxy. Ex. 6. Evaluate VCCî^^)^-

(Compare Exs. 4, 6, with Ex. 3.)

d^vEx. 6. Determine all the curves for which " ^ = 0.

dx^

Ex. 7. Find the curve at each of whose points the second derivative of

the ordinate with respect to the abscissa is four times the abscissa, and

which passes through the origin and the point (2,4) ..

Ex. 8. Find Prf'^^C^Ô^-x. 9. O f^psin d^)8.

GO. Successive integration with respect to two or more indepen-ent

variables. In this article the notation commonly used in this

kind of integration will be described ; and, in preparation for the

next article, a few examples will be given so that the student

may become familiar with the notation.

Suppose that fi(x,, z)= \f(x, y, z)

dz, (1)



124 INTEGRAL CALCULUS [Ch. VIII.

the integration indicated in the second member being performed

as if X, y were constants. (Itwill be remembered that if Xy y,---,

are independent variables, differentiation of F(x, y, "..,)ith

respect to one of the variables, say a?, is performed as if the others

were constants.)hen, suppose that

/2(",, z)= J/i(",, 2;)

y, (2)

the integration now being performed as if a;^ 2; were constants.

Again, suppose that

M^y Vi 2)= J/2("," z)

dx, (3)

the integration being performed as if y, z were constants. Equa-ion

(2),y virtue of equation (1),an be put in the form

/2(",y 2)= JlJV(^'^ ^) A

^y 5 W

and equation (3),y virtue of (4),an be wi'itten,

/aCaJ,, z)=J IJ IJ/(^'^ ^) 2=^2I^. (5)

The bracketing in the second member of (5)indicates that the

differential coefficient,f(x,y, z),is integrated with respect to z ;

that the result of this integration is then integrated with respect

to y ; and that, finally,the result of the last integration is inte-rated

with respect to a?. In the notation usually adopted, the

second member of (5)is abbreviated by removing the brackets,

and the order of the variables with respect to which the integra-ions

are made, is indicated by the order of the respective differ-ntials

of the variables beginning at the right and going toward the

left.Thus, the abbreviated form of the second member of (5)is

ffffi^yy z)dx dy dz. (6)




This is a triple integral. Similarly, the double integral in the

second member of (4)is generally written,

Jff(^yyyZ)dydz.As to the integration signs, the first on the right is taken with

the first differential on the right, which is dz in (6)above, the

second sign from the right is taken with the second differential

from the right, the third sign from the right is taken with the

third differential from the right, and so on. It is well to note

this usage, because attention must be paid to it when limits of

integration are assigned to a;, y, z* In some of the examples

below, and often in practical problems, the limits for one variable

are functions of one or more of the other variables.

Ex.1. Evaluate

) )

T xy^dxdydz.

If /denote the integral,

J= rf [zYxy^dxdy ^( Cxy^dxdy

Ex.2. Evaluate ] \' xy^dxdy.

* The notation described above is not universally adopted, but it is the

one most frequently used.



126 INTEGRAL CALCULUS [Cn. VIII.

61. Application of successive integration to the measurement of

areas : rectangular coordinates. In this and the two following

articles, proV)lems are solved which show applications of succes-sive

integration. In some of the examples there may not be

any special advantage in resorting to double integration, for the

reason that a single integration may suffice. They are, however,

given to the student for the purpose of making him familiar with

an instrument for solution which may sometimes be the only one

possible. It will be found that the elements in the summations

which follow are infinitesimals of a higher order than those which

have been met with heretofore.

Fig. 81.

Ex. 1. Find the area included between the parabolas whose equations are

3?/2 = 25", and 5x2 = 9y.

The parabolas are VOP^ WOP. Their points of intersection, 0, P, are

(0,0), (3,5). Taking any point Q within the area as a vertex, construct




a rectangle whose sides are parallel to the axes of coordinates and are equal

to Ax, Ay..

Produce the sides which are parallel to the x-axis until they

meet the curves in X, ilf,G^ B, thus forming the strip LGBM^ and produce

ML to meet the y-axis in B. On LM construct the rectangle HM^ giving it

a width Ay.

x==RM

Area of the rectangle HM"

limit^^^^ ^ Ax Ay.

Both y and Ay remain unchanged throughout this summation. Now,

BL=^,and i?i"f=

3\|.

Hence, area HM= \ \^

dx\Ay (1)

'=-"

(W|-S)a. (^)

As Ay approaches zero, the rectangle HM approaches coincidence with

the infinitesimal strip GM, and, in the limit, HM coincides with GM, Also,

the area OLPMO is the limit of the sum of all the strips similar to LGBM

lying between 0 and P, when Ay is made to approach zero as a limit.

Therefore,

y=py_

area OLPMO =

limit^yô/^(3^'|^)^2/

= 5.

If the linear unit is.an inch, the answer is in square inches. On substi-uting,

for lsJ^--^^\m3) its value as shown by (1) and (2),there is

obtained,

area OLPMO = ^'J_',[^^']^y

=

"f^''dydx. (4)3y_

25



128 INTEGRAL CALCULUS [Ch. VIII.

The latter is the customary abbreviated form which indicates that the first

integration is made with respect to x between the limits

_iLand 3-\/?or

", and that the result obtained thereby is to be integrated with respect to y

between the limits 0 and 5. The element of area in (4),namely dydx^ is an

infinitesimal of the second order.

Another way of performing the double summation required in adding up

all of the elements of area like dy dx, may be described as follows. Sum all

of these elements that are in the vertical strip ST, and then sum all of the

vertical strips in OLPMO. In the firstsummation, x and dx do not change,

and the upper and lower limits of y are 5-v/-,^

respectively ; in the second

summation, the limits are the values of x at 0 and P, namely 0 and 3. This

double summation is indicated by the double integral

s:s:fxdy.

9

This, on evaluation, gives an area of 6 square units as before.

The area OLPMO might have been expressed in terms of single integrals.

For

OLPMO = OLPN- OMPN

Jo ^3 Jo 9

= 10 - 6 = 5.

Ex. 2. Solve Exs. 7-11, Art. 29, by this method.

62. Application of successive integration to the measurement of

volumes: rectangular coordinates. If the equation of a surface

is given in the form

the volume can usually be determined by means of three successive

integrations. In the particular case of solids of revolution, the

volume can be found by a single integration. This was shown

in Art. 30. In Art. 61 the element of area was dydx, the area

of an infinitesimal rectangle each side of which was an infini-esimal.

In the case now to be considered, the element of volume

will be the volume, dxdydz, of an infinitesimal parallelepiped

each of whose infinitesimal edges is parallel to one of the axes

of coordinates. This will be illustrated in Ex. 1.



61-62. J SUCCESSIVE INTEGRATION 129

Ex. 1. Find the volume of the ellipsoid whose equation is

a;2 y2 g2

Let 0-ABC be one eighth of the ellipsoid whose volume is required.

Then OA = a, OB = b, OC = c. Take IL an infinitesimal distance dx on

Pio. 82,

OX, and through /, L pass the planes HIJ, KLM perpendicular to OX

Take EF of an infinitesimal length dy on ML^ and complete the infinitesimal

rectangle EFGD. Through the lines DE, GF pass planes parallel to the

plane ZOX which intersect the curvilinear surface HJMK in the infinitesi-al

arcs BV, ST. Through a point D' on DB, DD' having an infinitesimal

length dz, pass a plane parallel to the plane XOY. The infinitesimal paral-elepiped

D'F^ whose volume is dxdydz, will be taken for the element of

volume. The solid 0-AOB is the limit of the sum of parallelepipeds of this

kind. This limit will now be determined.

First, the volume of the vertical rectangular columnBF

willbe found

by

adding together all the infinitesimal parallelepipeds such as D'F which are

included between DEFG and BSTV.



130 INTEGRAL CALCULUS [Ch. VII!.

Second, the volume of the slice HIJMLK will be found by adding together

all the infinitesunal rectangular columns like BF which are erected between

IL and JM.

Third, the volume of O-ABC will be found by adding together all the

infinitesimal slices like HIJMLK that lie between OCB and A,

In the addition of the infinitesimal parallelopipeds from DEFG to BST T,

z alone varies, and it varies from zero to DB.

Vol. BF = dydz. 0)

In the addition of the vertical columns from IL to MJ, y alone varies,

and it varies from zero to IJ,

,;Yo\,HIJMLK

=\f f(fe

dy\dx,lyioL. 0 J J

(2)

In the addition of the slices between OCB and A, x varies from zero

to OA.

x = OA , y = IjrM=DR~= UA I y " lJ H=DR ^

". Vol. O-ABC = f If ^dzdyidx.

Writing this in the usual manner.

x=OA y = TJz = DR

Vol.

0-ABC= f r J dxdydz.

(3)

(4)x = 0 y = 0 " = 0

If the coordinates of B are a:, y, z, it follows from the equation of the

surface that

DB = z,

At the point J, z = 0; and hence.

'-^"-g-J

Also, OA = a

Therefore (4)becomes

Vol. O-^JJOJo Jo Jo

(?xdy dz.



02-63.] SUCCESSIVE INTEGRATION 181

On making the integrations in their proper order, it is found that

Vol. 0-ABC = iirabc.

Hence, the volume of the whole ellipsoid = fnabc.

Note 1. The volume of an infinitesimal parallelopiped is an infinitesimal

of the third order, the volume of a vertical column is an infinitesimal of the

second order, and the volume of a slice is an infinitesimal of the firstorder.

NoTB 2. Equally well, the planes bounding an infinitesimal slice might

have been taken perpendicular to either OZ or 0 Y,

Note 3. On putting a = b = c, the volume of a sphere of radius a is

found to be I ira*.

Ex. 2. Find the volume of the ellipsoid given in Ex. 1 : (a) by taking

the infinitesimal slice at right angles to OT; (") by taking it at right angles

to OZ.

Ex. 8. Determine the volume of a sphere of radius a by the method of

this article.

Ex. 4. Find the volume bounded by the hyperbolic paraboloid " =^,

the xy-plane and the planes z = a, x = A, y = b, y = B.^

Ex. 6. Find the volume of the wedge cut from the cylinder x^-\-y^ = a^

by the plane z = 0, and the part of the plane z = x tan a for which z is

positive.

Ex. 6. Find the entire volume bounded by the surface

\a \b \c

Ex. 7. The center of a sphere of radius a is on the surface of a right

blinder the radius of whose base ii

cylinder intercepted by the sphere.

cylinder the radius of whose base is -. Find the volume of the part of the

63. Farther application of successive integration to the measure-ent

of volumes: polar coordinates. The illustration in this

article is given, because the use of polar coordinates in dealing

with solids is often advantageous. It will be necessary to employ

these coordinates in solving some of the problems in Arts. 77, 79.

Ex. 1. To find the volume of a sphere of radius a by means of polar

coordinates. Let a point O on the surface of the sphere be the pole, the tan-

INTEGRAL CALC. " 10






I of sphere z= ft-I

ftolume ( : sphere = f' f f "''"^^M^ "am0d0

tf=0 t "^=rOL.

r=

0J

J

= ?|?p r^cos"^ sin ^ (W (i0

3 Jo

Ex. 2. Find the volume of sphere of radius a by thi^ method, on letting

the XO Y plane pass through the center.



CHAPTER IX

FURTHER GEOMETRICAL APPLICATIONS. MEAN

VALUES

64. The calculus has already been employed for the derivation

of the equations of curves in Art. 32, for the determination of

the areas of curves in Art. 27, for the determination of volumes

of solids of revolution in Art. 30, and for the determination of

volumes of solids in a more general case, in Arts. 62, 63. Carte-ian

coordinates were used in all but the last of these applica-ions.

This chapter will consider the derivation of the equations

of curves and the measurement of areas in cases in which polar

coordinates are employed. Special cases in areas and volumes

are taken up in Arts. 68-70. The measurement of the lengths of

curves for both Cartesian and polar coordinates is considered in

Arts. 71, 72 ; and the measurement of surfaces is discussed in

Arts. 74, 75. The subjectf mean values is treated in the lasttwo articles of the chapter.

65. Derivation of the equations of curves in polar coordinates.

Let the equation of a curve in polar coordinates be

/(r,^)=0.

It is shown in the differential calculus that, if (r,$)be any point

on the curve,\f/

he angle between the radius vector and the tan-ent

at (r,0),and "^the angle that this tangent makes with the

dBinitialline,then tan \l/ r "

,

dr

134



64-66,] FURTHER GEOMETRICAL APPLICATIONS 186

the length of the polar subtangent

the length of the polar subnormal

__dr~de

Ex. 1. Find the curve in which the polar suhnormal is proportional to

(is times)the sine of the vectorial angle.

drIn this case, " = k sin $.

de

Using the differential form, dr = k sin $d$j

and integrating, r = c " k cos ^.

If c = If, r = K (1 " cos ^),

the equation of the cardioid.

Ex. 2. Find the curve in which the polar subtangent is proportional to

(isK times)the length of the radius vector.

Ex. 8. Find the curve in which the angle between the radius vector and

the tangent is n times the vectorial angle. What is the curve when n = 1 ?

when n = J ?

66. Areas of curves when polar coordinates are used: by single

integration. Let AB be an arc of the curve r=f($),and suppose

that angle ÔX=:a, angle BOL = fi. The area of AOB is re-quired.

Divide the angle AOB into n parts, each equal to A^; then

The sector AOB will thus be divided into n sectors, which have

equal angles at 0. Let POQ be one of these sectors. About 0

as a center, and with a radius equal to OP^ describe through P a

circular arc PPj, which intersects OQ in Pi ; and about the same

center 0 with OQ as a radius describe an arc QQi,which meets

OP in Qi. The area of the sector POQ is greater than the area

of the "interior'' sector POPi, and is less than the area of the



136 INTEGRAL CALCULUS [Ch. IX.

"

exterior"

one QOQi. About 0 as a center, and through each of

the points in which the arc AB is intersected by the lines that

divide the angle AOB into equal parts, let circular arcs be drawn

which intersect the adjacentlines of division on each side, as

Fio. 84.

shown in Fig. 34. There is thus obtained a set of exterior circu-ar

sectors like QiOQ, and a set of interior circular sectors like

POPy. It has been seen that each sector POQ of the figure AOB

is gi*eater than the corresponding interior sector POPi, and less

than the corresponding exterior sector QOQi ; that is,

sector POPi " sector POQ " sector QiOQ.

Therefore the sum of the sectors POQ, which is AOB, is

greater than the sum of the interior sectors, and less than the

sum of the exterior sectors ; that is,

^POPiAOB

"^QOQ,.



66.] FURTHER GEOMETRICAL APPLICATIONS 137

In the limiting case, when the number of sectors POQ becomes

infinite,that is when A$ approaches zero, the sum of the areas of

the interior sectors, and the sum of the areas of the exterior sectors

approach equality. For, the difference between these two sums is

equal to the area of AMM'A\ which is ^(OM^ - 0A^)^$, and

can therefore be made as small as one pleases by decreasing A^.

The area AOB always lies between these sums ; and hence,

area AOB =

limit^^ôy^ÔPi.If the coordinates of P be denoted by r, $, the area of

POPi=ir'Ae.

Hence, are'a AOB = limit^^

=0/

i ^^^ 5 ".

that is, area AOB =i fVde,

by the definition of a definite integral. The element of area for

polar coordinates is thus ^ r^dO,

Ex. 1. Find the area of the sector of the logarithmic spiral whose equation

is r = c"^, between the radii vectores for which ^ = o, tf= /3.

Area POQ = U^f^dB

4a

ri, r2 being the bounding radii

vectores.F16. 86w

Ex. 2. Find the area of one loop of the lemniscate r^ = a^C082 ^.

The area of one half the loop,^

'

OMA =

i(f^de,

between proper limits for ^, which must be determined.




The initial and final positions of the radius vector are OA and OL the tan-ent

to the arc OM at 0. For r = 0, the equation of the curve givês

0 = aaco82^;

and hence,

Fig. 86.

The positive sign indicates the position of OL, and the negative sign that

of ON.

Hence, area OMA

-^S!^de

Hence,

= "f^cos2^d^

2 Jo

4*

area of loop OMARO = " .

2

Ex. 8. Find the area of a sector of the spiral of Archimedes, r = a0,

between $ = a, 0 = p,

Ex. 4. Find the area of the part of the parabola r=â sec^ - intercepted

between the curve and the latus rectum.2

Ex. 6. Find the area of the cardioid r^ = a^ cos -"

2

Ex. 6. Find the area of the loop of the folium of Descartes,

ajs + ys _ 3 fjrg^y. 0.

(Hint: Change to polar co5rdinates, thus obtaining r =

3 g sin ^ cos g

cos* e + sin* e; and

then change the variable B by putting z = tan $.)

Ex. 7. Show that the area bounded by any two radii vectores of the

hyperbolic spiral 7*8 = a, is proportional to the difference between the lengths

of these radii.

Ex. 8. Show that the area of a loop of the curve r^ = a^ cos n$ is " .

n



66-67.] FURTHER GEOMETRICAL APPLICATIONS 139

67. Areas of curves when polar coordinatesare used: by double

integration. The areas of curves whose equations are given in

polar coordinatescan be found by double integration,n a man-er

analogous to that used in Art. 61. Example 1 below will

serve to make the method plain. Successiveintegrationin two

variables,polar coordinates,willalsobe required in Arts. 77, 79.

I

Ex. 1. Find by doable integrationthe area of the circlewhose equation

is r = 2acos^.

Let OLAN be the given circle,0

being the pole and OA the diameter

2 a. Within the circle take any

point P with coordinates (r,0),Draw OP and produce ita distance

Ar to S, Revolve the lineOPS about

0 through an angleAS to the position

OQR^ Then

area PQBS= i[(""^*')*^} ^^

z=:rArAe'î(ÂryA9.Fig. 87.

Produce OP, OQ to meet the circlein Jf, G. The area of the sector MOG

will be found by adding allthe elements PQBS therein,and the area of the

semicircle OLMA will be calculated by adding all the sectors like MOG

that itcontains.

Area.MOG = 2iPQRS

r=ajr

=

\imitj:^r=o/](rArAeJ(Ar)"^)r = 0

r=OJf

= limit^= 0 / rArAB

r=ii

by a fundamental theorem in the calculus,*

r=OM

and hence, area MOG = irdrAd,

* See Note D, Appendix.



140 INTEGRAL CALCULUS [Ch.IX.

in which the integrationisperformed with respect to r. Hence,

B=TOX r=OM

area OLMA = f Tfr dr IcW"=0 r=0

Jo Jo

2

Hence, the area of the circleiswa^.

The area of OMA can alsobe obtained by findingthe area of the circular

stripLG whose arcs are distantr, r + dr from 0, and then adding allof the

similarconcentriccircularstripsfrom O to A. The angle LOG = cos"^ ^" It

willbe found that

JO

"^---2

2a

area OMAJo Jo

'

rdrd0 = ^ as before.

Ex. 2. Find by double integrationthe area of the circleof radius o, the

pole being at the center ; (1)by adding equiangularsectors ; (2)by adding

concentric circularstrips.

I

68. Areas in Cartesian coordinateswith oblique axes. In this

case the method of findingthe area issimilarto that in Art. 61.

Let the axes be inclinedat

an angle cu, and construct a

parallelogram whose sides

are parallelto the axes and

have the lengthsAa5,

Ay.The area of this parallelo-ram

is

Ax Ay sino).

The whole area is the limit

of the sum of all parallelo-rams

that are constructed within the perimeter of the figure

when

Ax

and

Ay approach zero. Hence, the area is the value of

the double integral

IIsin0) c2a;y, that is,sina" | idx dy,

Fie. 88.






When 35 = 0, ^ = 0; and when " = 2ira, ^ = 2ir.

Also, (to = o (1 " cos 0)dd.

If these values fory, dx, and the limits, be substituted in the integral

above, it becomes

area =

a2f''(I

cos^)"d^

=

a2j^''(l-2cosî"f^)(W

That is,the area is three times that of the generating circle.

Ex. 8. Find the area of the ellipse -3 +p

= 1- (Compare Ex. 3, Art. 27.)

(Hint: put x = acos0, then y = 6 sin0.)

Ex. 4. Find the volume of the solid generated by the revolution of a

complete arch of the cycloid of Ex. 2 about the x-axis.

70. Measurement of the volumes of solids by means of infinitely

thin cross-sections. In Art. 30 the volume of a solid of revolution

was determined by finding the volume of an infinitely thin slice

of the solid,the slice being taken at right angles to the axis of the

figure, and the sum of the volumes of all such slices being then

found. This method can be extended to other figures besides

figures of revolution. Some convenient line is chosen, and an

infinitesimally thin slice of the solid is taken at right angles to

this line. If the area of a face of the thin slice can be expressed

in terms of its distance from some point on the line, the volume

of the slice can be expressed in terms of this distance ; and from

this, the sum of the volumes of all the slices can be found. For

example, let the chosen line be taken for the axis of a?, and sup-ose

that the area of a face of a thin slice at right angles to this

line is /("). Let the thickness of the slice be Aa?. The volume

is (asin Art. 30) the limit of the sum of an infinite number of

infinitesimal cylinders whose volumes are of the form /(a?)a5.

That is,




voluiDe of the solid = limitsô

S f(x)

=Jf(x)dx,in which the limits of integration are determined from the figure.

Ex. 1. Determine by this method the volume of the ellipsoid

(The studentis

advisedto

makea figure.

)At a distance x from the

center

cut out a very thin slice at right angles to the x-axis, and let its thickness be

dx. The face of this cylindrical slice will be an ellipse whose semiaxes are

These values are deduced from the equation of the ellipsoid.

The area

ofthis

ellipse=

irftc/l -

^V

Hence, the volume of the slice =irbc[1 " ^ j

dx ;

and therefore, volume of ellipsoid = irbci *(l")d!K

= I wobc.

Ex. 2. Find the volume of a sphere of radius a by this method.

Ex. 8. Find the volume of the torus generated by revolving about the

oâxis the circle x^ + (y " b)^= a^, in which 6 " a.

Ex. 4. Find the volume of a pyramid or a done having a base B and a

height h.

Ex. 5. Find the volume of a right conoid with a circular base and alti-ude

h, the radius of the base being a.

Ex. 6. A rectangle moves from a fixed point, one side varying as the

distance from this point, and the other as the square of this distance. At

the distance of 2 feet, the rectangle becomes a square of 3 feet. What is

the volume then generated ?

Ex. 7. Given a right cylinder of altitude h, and radius of base a.

Through a diameter of the upper base two planes are i"assed touching the

lower base on opposite sides. Find the volume included between the planes.

Ex. 8. Find the volume of the elliptic paraboloid 2 x =^

+^

cut off by

the plane x = h, P Q




71. Lengths of curves: rectangular coordinates. To find the

length of a curve is equivalent to finding the straight line that

has the same length as the curve. For this reason the measure-ment

of its length is usually called **the rectification of the

curve."* The deduction here made of the integration formulae

for the length of a curve depends

upon the definition that integra-ion

is a process of summation.

The equation of a given curve

is /(",y)= 0 ; it is required to

find the length a of an arc AB,

A being the point (xj,yi),be-ng

the point (ajj,2)-On the

curve take any two points P, Q

whose coordinates are ", y, and

X-f

Aoj, y -f^y. Draw the chord PQ and make the construction

indicated in the figure.

The chord PQ = V(Aaj)*-f (Ay)* (1)

(2)

As Q approaches infinitely near to P, that is, when Aa? ap-roache

zero, the chord PQ approaches coincidence with the

arc PQ. It is shown in the differential calculus that if Aa? is

an infinitesimal of the first order, the difference between the

* In 1659 Wallis (see footnote, Art. 27) published a tract in which he

showed a method by which curves could be rectified, and in 1660 one of his

pupils, William Neil, found the length of an arc of the semi-cubical parabola

01?= aiP.This is the firstcurve that was rectified. Before this it had been

generally supposed that no curve could be measured by a mathematical proc-ss.

The second curve whose length was found is the cycloid. Its rectifi-ation

was

effectedby Sir Christk)pher Wren

(1632-1723)and publishedin

1673. This was before the development of the calculus by Leibniz and

Newton.




arc and its chord is an infinitesimal of at least the third order;

that is,

arc PQ=

chord PQ-^h, (3)

in which iîs an infinitesimal of the third order when Ao? is an

infinitesimal of the firstorder.

Therefore,s=S(arcsPQ)

.lMit^.,-"yh^'^by a fundamental theorem.* As Ax approaches zero, "

^ in gen-Ax

eral approaches a definite limiting value, namely, --^.Therefore,

by the definition of a definite integral.

In applying this formula it will be necessary to express

^l+(-^]n terms of x before integration is attempted.

Instead of being put in the form (2),equation (1)may be

given the form,

chord PQ

=yJlf^jAy.

By the same reasoning as above, it can then be shown that

in which -i/l ( ) must be expressed in terms of y before inte-ration

is performed. Formula (4)or formula (6)will be used,





according as it is more convenient to take x or y for the inde-endent

variable.

If A" denotes the length of the arc PQ, it follows from (3)that

Therefore, by the differentialcalculus.

t-yHM-

whence, ds

=\/l+( 3^ ]^"

Similarly,

ds^yjl-hl^)

y:

In order to recall formulae (4)and (5)immediately whenever

they may happen to be required, the student need only remember

the construction of the triangle PQRy and let its sides become

infinitesimal.

Ex. 1. Find the length

ofthe

circle whose equationis 35^

+ y*= a^.

Let AB be the firstquadiantal arc of the circle.

In this case,S^

= _

?.

dx y

m^ "" ^ = f-y/i(!)"*/.'a/m^*dx

0 Va'^ - x3

2*

Therefore, the perimeter of the circle (=4 AB) = 2 ra.

Ex. 2.Find the

lengthof the arc of the parabola from the vertex to the

point (xu Vi). Find the length of the arc from the vertex to the end of the

latus rectum.




Ex. 8. Find the length of the arc of the semicubical parabola atf^ x^

from the origin to the point (xi,yi). Also to the point for which x = 5 a.

Ex. 4. Find the lengthof

the arc

ofthe

catenary y=

^("*fe

"}from

the vertex to the point (xi,y{). Also to the point for which x = a.

Ex. 6. Find the length of the arc of the cycloid from the point at which

^ = ^0 to the point at which 9z=e\. Also find the length of a complete arch

of the curve.

Ex. 6. Find the entire length of the hypocycloid x* -

Ex. 7. Show that inthe ellipse

x = asin0, y = 6co8^,

"pbeing the complement of the eccentric angle of the point (x,y),the arc "

measured from the extremity of the minor axis is

+ y* =

a*.

Jo

- c2 sin* 0 c?0,

and that the entire length of the ellipse is

rl ,

4a I Vl-easin2 0"i0,Jo

in which e is the eccentricity.*

72. Lengths of curves: polar coordinates. The equation of a

curve is /(r,$)" 0, and the length s of the are AB is required, A

being the point (ri,i),and B the

point (rj,2)- On the curve take anytwo points P, Q, whose coordinates

are r,. 6, r + Ar, ^ + A^. Draw OP,

OQ, and the chord FQ. About 0 as

a center, and with a radius equal toy X JIp

OP, describe the arc PR which inter-ects

OQ injK,and

draw PRi at right

angles to OQ. Then the angle

POQ = ^0, RQ = Ar,^-""^ )i

---^

and arc o^^"^ \ ^'j x_

^" .,

Fig. 40.

PR = rA^.

"This integral, which is known as **the elliptic integral of the second

kind/* cannot be expressed, in a finiteform, in terms of the ordinary func-ions

of mathematics. See Ex. 8, Art. 83.

INTEGRAL CALC. " 11




It is shown in the differential calculus that when A^ is an infini-esimal

of the firstorder,

PRi = arc PE " ig, an infinitesimal of the third order ;

QEi = QR 4" t'a, an infinitesimal of the second order;

chord FQ = arc PQ " 13, an infinitesimal of the third order.

In the right-angled triangle PBiQ,

chord PQ= VPiî*

-f BiQ^.

Hence, when A^ approaches zero.

arc PQ- i's= V(PB-isy-h{BQ-î2y,

or, arc PQ = V(rAd -

hY + (Ar + hf -f*'" (1)

^J^+f^y^^llh^2i,--^''-^''

A^ + i'^ (2)\ ^VA^y A^^ ^(Ad)"^ (A^)"

^ ^'^ ^

which differs by an infinitesimal of at least the second order from

M^)2

Aft

Therefore,

s = SPQ = limit^,.oy^+(|^YA^,

when A^ approaches zero, -" approaches the definite limiting

dr

value "

.Therefore, by the definition of a definite integral,

"=j;'ARif'"-"""

It will be necessary to expressx/^+f-^)

^ terms of $ before

integration is made.^

^ ".t

Similarly, on removing Ar from the radiciai sign in (1),t can

beshown

that



72-73.] FUBTHEB GEOMETRICAL APPLICATIONS 149

in which -v/l-r^( )must be expressed in terms of r before inte-ration

is made. 1 ormula (3)or (4)will be used according as it

is more convenient to take d or r for the independent variable.

In order to recall these formulae immediately it is only

necessary for the student to remember the construction of the

figure PBQ, and to suppose that its sides are infinitesimal.

Ex. 1. Find the length of the circumference of the circle whose equa-ion

is

Here ^= 0.

d9

Hence.

-r"/^(SF'**=

"/"= 2 ira.

Ex. 2. Find the length of the circle of which the equation is r = 2 a sin 0.

Ex. 3. Find the entire length of the cardioid, r = a(l"

cos^).

Ex. 4. Find the arc of the spiralof Archimedes, r = aO, between the points

(n, î),(r2, 2).

Ex. 6. Find the length of the hyperbolic spiral, r0 = a, from (n, î)to

(r2, 2).

Ex. 6. Find the length of the logarithmic spiral, r = e^, from (1,0")to

(ri, 1).

Ex. 7. Find the length of the arc of the cissoid r = 2a tan ^ sin 0 from the

cusp (^ = 0) to^=-.4

Ex. 8. Find the length of the arc of the parabola r = q sec^ - from ^ = 0

_ _

r 2

to 0 = 0i; also, from ^ =

--to^= -.

2 2

..":";;;:"

73. The intrinsic 'equation of a curve. Let PQ be the arc of a

given curve, and let s denote its length. Suppose a point starts

at P and moves along the curve towards Q. At the instant



150 INTEGRAL CALCULUS [Ch.rx.

of starting the point moves in the direction of the tangent

PTi. In passing over the arc PQ the direction of motion

changes at every instant, until at Q the point is moving in

the direction of the tangent QT^. The total change in direc-ion,

as the point moves from P to Q, is measured by the

angle ^ between the two tangents. It will be found that a rela-ion

exists between the distance s through which the point has

moved, and the angle ^ by which the direction of its motion has

changed. This relation between s and ^ is called the intrinsic

^r,equation of the curve. The form

of this equation depends only on

the nature of the curve, and the

choice of the initialpoint P. On

the other hand, the form of the

equation of a curve in other

systems of coordinates, for ex-ample

the rectangular and polar,

depends upon points and lines

that are independent of the curve.

Hence the term " intrinsic.''

To find the intrinsic equation of a curve given in rectangular

or polar coordinates,

(1)Determine the length of arc 8 measured from some con-venient

starting point up to a variable point on the curve.

(2)Find the angle ^ between the tangents at the initial and

the terminal points.

(3)Eliminate the

rectangularor

polar variablesfrom the

equa-ionsthus found.

Fig. 41.

Ex. 1. Find the intrinsic eqaation of the catenary

If the vertex of the curve be taken as starting point,

(1) " = ^(e- e "). [Ex.4, Art. 71.]2




Also, since the tangent at the vertex is parallel with the as-azis,

(2)

tan^=gi(e--e"-).

The elimination of x from (1)and (2)gives the required equation between

s and 0, viz.,

8 = a tan ^.

It is easy to extend this result and show that

8 = a [tan("f" "f"i)tan 4"i]

is the intrinsic equation of the catenary when any point A is chosen for the

initial point. The angle î is then the angle between the tangent at the

vertex and the tangent at the point A.

Ex. 2. Find the intrinsic equation of the parabola

r= a sec^-.

2

If the vertex of the

parabola

be taken for the initialpoint,

(1) a =

atan?sec^+alogtan/|?) [Ex. 8, Art. 72.]

Also, since the tangent at the vertex makes an angle of - with the polar

axis,^

where 4"'is the angle that the tangent at the point (r,0) makes with the

polar axis. But

2 0' = ^ + T.

Hence, ^ = 2^.

On substituting this value of ^ in equation (1) the intrinsic equation of

the parabola is found to be

" = a tan 0 sec ^ + a log tanf^

+ - j

EXAMPLES.

1. Find the intrinsic equation of a circle with radius r.

2. Find the intrinsic equation of the cardioid r =

a(l"

cos^),the arc

being measured from the polar origin.

8. Find the intrinsic equation of the cycloid

x =

a(e"

sin^),'

y-

) =

a(^"

8in^),lr =

a(l"

cosd),/




(1) the origin being the initialpoint,

(2) the vertex being the initialpoint.

4. Find the intrinsic equation of the parabola y* = 4jm5,

(1) the vertex being the initialpoint,

(2) the extremity of the latus rectum being the initialpoint.

5. Find the intrinsic equation of the semicubical parabola 3 ay'^= 2 2B*,

taking the origin for initialpoint.

"E6. Find the intrinsic equation of the curve yâ log sec -, taking the

origin for the initialpoint.

7. Find the intrinsic equation of the logarithmic spiral r = a"^"

8. Find the intrinsic equation of the tractrix

y

taking the point (0,c) as the initialpoint.

9. Find the intrinsic equation of the hypocycloid x' + y' = a', taking

anyone

of

the cusps as initial

point.

74. Areas of surfaces of solids of revolution. Suppose that the

surface is generated by the revolution about the a?-axis of the

arc AB of the curve whose equation is y =/(") ; and let the co5r-

R T L s

Fig. 42.

dinates of the points A, B, be a?,, 3/1,and x^, y2" respectively.

Take (Fig.42) any two points on the curve, say P, Q, whose

coordinates are x, y, and x-fAa?, y-j-Ay. Draw the chord PQ




Fio. 48.

and the ordinates MP, SQ,

and suppose that LM is an

ordinate which is not less,and

that TN is an ordinate which

is not-greater than any ordi-ate

that can be drawn from

the SiTcPNMQ to the a-axis.

(In Fig. 43, LM coincides

with SQy and TN coincides

with RP.) Through N, M,

draw lines PiQ\,P^Qifparallel

to the ovaxis and equal in

length to the arc PNMQ, On the revolution of the arc AB

about OXj each point in AB describes a circle with its ordinate

as radius.

The surface generated by arc PNMQ

^2'jrLMX2ilCQPQy.

and ^ 2irTN x arc PQ.

When Ax is an infinitesimal of the firstorder,

arc

PQ=

chord PQ +tj, an infinitesimal

of at

least the thirdorder;

LM= BP-^-iy an infinitesimal of at least the firstorder;

TN^ RP"i\ an infinitesimal of at least the firstorder.

Hence, since the chord PQ

=vÂx, it follows that

^ surface generated by arc PQ

i+(g)'..+,)

Therefore,




^ surface generated by AB

"

"=*! :

liiniW-0^2y + i\(\1+(f|Y-+

^)-

By a fundamental theorem* the least and the greatest expres-ions

in this inequality are each equal to

" = *!,

Hence, surface generated by AB

* = *!

=

Hmit:^=o^2ryyjl^f^'^.

When Ax approaches zero,-^

takes a definite limiting value,

dv^^

namely -^"Therefore, by the definition of a definite integral,

ax

area I of surface

=J^^*2"yVl(||)*^^'(1)

It is necessary to express the function under the sign of inte-ration

in terms of x before integration is performed. If

be used for the length of the chord, there will result,

"r"

area of surface

=J^^*2iry^H-(^)*"l2)

Formula (1)or formula (2)is taken according as it is more

convenient to choose x ov y for the independent variable.

The surface generated by the revolution of AB about the y-oxis

is

givenby the formulae,




74.] FURTHEB GEOMETRICAL APPLICATIONS 166

s"rfaee=r2^Vl^^^-

(8)

(4)

The student is advised to deduce these formulae for himself.

The expressions under the sign of integration in formulae (1),

(2)may both be written 2wy

ds hy Art. 71, and those in formulae

(8),(4)may both be written 2 wx ds. In order to recall immedi-tely

a foi-mula for the area of a surface of revolution, it is only

necessary to remember that the area traced out by an infinitesi-al

arc in its revolution about any line is equal to the product of

the length of the infinitesimal arc by the length of the circle

which is described by a point on the arc.

Ex. 1. Find the surface generated by the revolution of a

semicircle ofradius a about its diameter.

Let the diameter be the x-axis,

and the origin be at the center ;

the equation of the curve will be

"" + y* = o*.

Surface generated by ABA'

about a;-axis

'^-rnMir*-But, 1 +

Hence,

""= 1 +fl;2

_

g^ + y'_

q'

surface = 2 xa j dx

Ex. d. Find the surface of the prolate spheroid obtained by revolving

about the oc-axisthe ellipse h^^-\-aV

= "^"^-

Ex., 3. Find the surface generated by revolving about the a;-axis the

parabola y*= 4

ax.Show

that the curved surface ofthe figure

generatedby the arc between the vertex and the latus rectum is 1.219 times the area

of its base.




Ex. 4. Find the surface generated by revolving about the y-axis the

catenary y=-fe"-fe "]from x = 0 to x = a.

Ex. 6. Find the entire surface generated by revolving about the x-axis

the hypocycloid x' + y"

= a*

Ex. 6. A quadrant of a circle of radius a revolves about the tangent at

one extremity. Find the area of the curved surface generated.

Ex, 7. The cardioid r = a (1 + cos e) revolves ab"Tut the initial line.

Find the area of the surface generated.

75. Areas of surfaces whose equations liave the form z^fipc^y).

Areas of surfaces of revolution were considered in the last article.

A more general case will now be discussed. In the explanation

of the following method for measuring the area of a surface,

referencewill be made to these two geometrical

propositions:

(a) The area of the orthogonal projectionf a plane area upon

a second plane is equal to the area of the portion projectedulti-lied

by the cosine of the angle between the planes. (SeeC.

Smith, Solid Geometry, Art. 31.)

(b) If the equation of a surface be in the form z

=/(", y),the

cosine of the angle between the ipy-plane and the tangent plane at

any point {x,y, z)of the surface is

MtH%)V-(SeeC. Smith, Solid Geometry, Arts. 206, 26.)

Let z = f(x,y)be the equation of the surface BFCMALB

whose area is required. Take two points P, Q,whose coordinates

are x, y,z,X'{- Ax, y -{-Ay, z

-{-Az, respectively. Through P and

Q pass planes parallel to the ^2;-plane and let them intersect the

surface in the arcs ML, M^Ly Also pass planes through P, Q,

parallel to the gaj-plane. The curvilinear figure PQ is thus

formed. Theprojectionf the surface PQ on the i"y-plane is the

rectangle PiQi whose area is Ax Ay. When Ax, Ay approach zero,

the point Q comes infinitely close to P; and the curvilinear sur-




face PQ, which is then infinitesimal, approaches coincidence with

thai portion of the tangent plane at P, which also has PxQi ior its

projectionn the a^-plane. The area of PjQi also becomes dxdy.

Fio. 46.

Now let Q be infinitelynear to P. Ify

is the angle between the

a?y-plane and the tangent plane at P, it follows from (a)and the

remarks whichhave justbeen made, that

Hence,

area PiQi = area PQ " cos y.

area PQ = area PjQi " sec y

= dx dy sec y.

Therefore, by (6),area PQ =

yjl^f^\f^Yx dy.

The summation of all the infinitesimal surfaces PQ in the strip

LMMiLi gives

area of strip LM. =

[/ i+ (:")X^hr



158 INTEOBAL CALCULUS [Ch. IX.

The summation of all the strips like LM^ in the surface

BFCMALB gives

area"

" " --"-- -

CSS 01 f^SL I

, of sxMace BFCMALB=C

[ C

\^+(^'+(j-)

or, abbreviating in the usual way,

The limits y = SL, x = OA can be determined from the equa-ion

of the surface. It is necessary to express the function under

the signs of integration in terms of x and y. It may happen that

a more convenient form of the equation of the surface is either

x = f(y,z),

r y

=f(z,x).

The area of the surface will then be

the value of either one or the other of the double integrals

between the proper limits of integration.

In some cases,

thereare two

surfaces each of whichintercepts

a portion of the other. In finding the area of the intercepted

portion of one of the surfaces, it is necessary to obtain the partial

derivatives that are required in the formulae of integration, from

the equation of the surface whose partial area is being sought.

This is illustrated in Ex. 2.

Ex. 1. Find the surface of the sphere whose equation is

xs + ys + "2 = a^.

Let 0-ABC (Fig.46) be one eighth of the sphere. In this case,

dx z' dy "*

\dx) \dy/ z^ z^ z^ a2-x2-ya




XTÔA y=8L

Therefore, area of surface ^5C = J J î+l^\\l^y(ix

Jo Jo Va^ - X2 _ y2

^0 L Vo^ " x-'-ô

2 Jo 2

Hence, area of all the surface of the sphere is 4 ra*. (Compare Ex. 1,

Art. 74.)

Ex. 2. The center of a sphere, whose radius is a, is on the surface of a

right cylinder the radius of whose base is i a. Find the surface of the cylin-er

intercepted by the sphere. On taking the origin at the center of the

sphere, an element of the cylinder for the 2r-axisand a diameter of a right

section of the cylinder for the a"-axis,the equation of the sphere will be

and the equation of the cylinder, x^ + t/^ ax.




The area of the strip GP will first be found, and then the strips in the

cylindrical surface APBOCA will be summed. The element of surface in

the strip CP iadzdz. Hence,

Cylindrical surface intercepted = 4 APBOCA

Since the surface required is on the cylinder, the partial derivatives must

be derived from the second of the equations above. Hence,

dy_a-2x

djL^Q

dx 2y'

dz

Also, CP^ = "a = a* " (x + y2),since P is on the sphere,

and hence, =a^ " ax, since P is on the cylinder.

Moreover, OA = a.

Therefore,the cylindrical surface

intercepted

-if L'-(^")']**-But on the cylinder, y'^= ax " x^. Hence,

the intercepted cylindrical surface

^"'-'"

dxdz

=2a((

^0 Jo Vox - x*

=

2ar^^^^dx= 2arJ^-dx

"^0 Vax - x2 -^0^x

= 4a2.

Ex. 3. In thepreceding example,

find thesurface of

thesphere

inter-epted

by the cylinder.

Ex. 4. Find the area of the portion of the surface of the sphere

x2 + y2 + 2ra- 2 ay

lying within the paraboloid

y = Ax^-\- BzK

76. Mean values. The mean value of n quantities is the nth

part of their sum. Let 4^(x)be any continuous function of a?,




and let an interval, 6 " a, be divided into n parts, each equal to h.

The mean value of the n quantities,

"^(a),^(a+ ^),it"{a 2h),

...,

"^(a+ (n- 1)^),

"\"{a)'\-"f"{a-\-h)'\-"\"{a-\-2h)-\"^(a4-n 1^)

n

Since n = "7^,this mean value is

6 " a

Now suppose that x takes all the possible values, infinite in

number, that are in the interval between a and 6. Then, n is

infinite,h is infinitesimal, and the number of terms in the last

numerator is infinite. The sum of all these terms, by Art. 4, is

expressed by

fj"(x)dx.

Hence, the mean value of all the values that a continuous

function, ^(a?),an take in the interval 6 " a for a? is

J^4"(ag)clag

This is usually called the mean value of the function"\"(x)

ver

the range b " a. A geometric conception of the mean value was

given in Art. 7 (a). A more general definition of mean value is

given in Art. 77.

It is necessary to understand clearly the law according to

which the successive values of the function are taken. Exs. 1, 2,

Exs. 6, 7, and Exs. 12, 13, will serve to illustrate this remark.

Ex. 1. Find the mean velocity of a body when falling from rest, the

velocitiesbeing taken at

equalintervals

oftime.

In the case of a body falling from rest, v = igt. Hence, calling V the

mean velocity for a time tu




\vdt

that is, the mean velocity is one half the final velocity.

Ex. 8. In the case of a body that falls from rest, find the mean of the

velocities which the body has at equal intervals of space. It is known that,

if 8 is the distance through which a body has fallen on starting from rest,

and V is the velocity required.

Hence, the mean velocity,

that is, the mean of the equal-distance velocities is equal to two thirds of the

finalVelocity..

Ex. 8. Findthe average value of the

function3a;3 +

5x "

7as x varies

continuously from 1 to 4.

Ex. 4. Find the average value of the function "^ " 3aj" + 2a;^l ass

varies continuously from 0 to 3.

Ex. 5. Find the average ordinate drawn,

(o) in the curve, y = x^ + x + 1 between the abscissas 2, 3 ;

(6) in the curve, y = (x+ l)(x + 2) between the abscissas 1, 3 ;

(c) in the curve, x* + ox* + cM^ + ft^y= o between the abscissas o, 0.

Ex. 6. Find the mean length of the ordinates of a semicircle (radiusa),the ordinates being erected at equidistant intervals on the diameter.

Ex. 7. Find the mean length of the ordinates of a semicircle (radiusa),the ordinates being drawn at equidistant intervals on the arc.

Ex. 8. Find the mean value of sin ^ as ^ varies from 0 to -"

2

Ex. 9. Find the mean distance of the points on the circumference of a

circle of radius a, from a fixed point on the circumference.

Ex. 10. Find the mean latitude of all the places north of the equator.

Ex. 11. A number n is divided at random into two parts. Find the mean

value of their product.

Ex. 12. Show that the mean

ofthe

squareson the diameters

ofan

ellipsethat are drawn at equal angular intervals is equal to the rectangle contained

by the major and minor axes.



76-77.] FURTBEB GEOMETRICAL APPLICATIONS 163

Ex. 18. Show that the mean of the squares on the diameters of an ellipse

that are drawn at points on the curve whose eccentric angles differ succes"

sively by equal amounts, is equal to one half the sum of the squares on the

major and minor axes.

77. A more general definition of mean value. In Exs. 1, 3, 4,

below, " the range"

over which the function (inthese cases, the

distance of a point)varies,is a plane area. In Ex. 2, the range

is a curvilinear area ; and in Exs. 5, 6, it is a portion of space.

The following may be taken for the definition of the mean value

of a function, whatever the range may be :.

The mean value of a

fimction throughout

any range

(The value of the ftinctionfor eaeh

element of the range) x (the ele-

.mentof the range)

The range

in which the summation in the numerator is made throughout

the whole of the range. The mean value considered in Art. 76

is merely a special case.

Ex. 1. Find the mean distance of a fixed point on the circumference of a

circleof radius a from all points within the circle.

On taking the fixed point for the pole and the tangent thereat for the

initial line, the value of the function (inthis case the distance)t any point

(r,e) is r. The element of the range

(inthis case an area)at the point is

rdedr. This is shown in Fig. 47.

Hence,

the mean distance =

i I r^drde

_

Jo Jo

Jo Jordrde

8a8

^

1 sin8 e'de3 Jo

9ir

(SeeEx. 1, Art. 67.)

INTEGBAL CALC. " 12




Ex. 8. Find the mean length of the ordinates drawn from all points on

the curved surface of a hemisphere of radius a to itsdiametral plane.

Ex. 8. Find the mean length of the ordinates drawn from all the points of

its diametral plane to the surface of the hemisphere of radius a.

Ex. 4. Find the mean square of the distance of a point within a given

square (side 2a)

from the center of the square.

Ex. 5. Fmd the mean distance of all the points within a sphere of radius a

from a given point on the surface.

Ex. 6. Find the mean distanceof alltbe points within

a

sphere of radiusa

from the center.

EXAMPLES ON CHAPTER IX.

1. Find the volume of a sphere of radius a by means of a single integra-ion.

(Suppose that the sphere is made up of infinitely thin concentric

spherical shells of thickness dr. The volume of each shell = 4TrMr ; hence

volume of sphere = 4 ir I n^dr = J wa^.

Jo

8. Find the volume and surface generated by revolving about the y-axis

the ellipse 6%2 + ^2^2-

^252,

8. Find the surface generated by the revolution about the ysuasoi the arc

of the parabola y^ = 4ax from the origin to the point (",y),

8 a?4. Find the volume generated by revolving the witch y = " " " - about

its asymptote.^

6. Find the convex surface of the cone generated by revolving about the

ic-axisthe line joiningthe origin and the point (a,b).

6. Find the surface of the torus generated by revolving about the a;-axis

the circle x^ +(y - 6)2 = a*.

7. On the double ordinates of the ellipse ^-f^= 1, and in planes per-

a2 52

pendicular to that of the ellipse, isosceles triangles of vertical angle 2 a are

described. Find the volume of the surface thus constructed.

8. Two cylinders of equal altitude h have a circle of radius a for their

common upper base. Their lower bases are tangent to each other. Find the

volume common to the two cylinders.

9. Find the volume inclosed by two right circular cylinders of equal

radius a whose axes intersect at right angles. Alaa^ find the. surface of one

intercepted by the other.



77.] FUBTHER GEOMETRICAL APPLICATIONS 165

10. Find the volume of the solid contained between

the paraboloid of revolution, x^

-\-ff^âz)

the cylinder, "? + y^ = 2 ax ;

and the plane, ;? = 0.

11. Find the entire volume bounded by the surface

"' + y* 4- "* = o*

12. An arc of a circle revolves about its chord. Find the volume and sur-face

of the solid generated, a being the radius, and 2 a the angular measure of

the arc.

18. A cycloid revolves about the tangent at the vertex. Find the surface

and volume of the solid generated.

14. A cycloid revolves about its base. Find the area of the surface

generated.

15. A cycloid revolves about its axis. Find the surface and volume

generated.

16. A quadrant of an ellipse revolves round a tangent at the end of the

minor axis of the ellipse. Find the volume of the solid generated.

17. If 6 be the radius of the middle section of a cask, a the radius of

either end, and h itslength, find the volume of the cask, assuming that the

generating curve is an arc of a parabola.

18. Find the length of the curve âif^ z{x" Z

o)2from a; = 0tox = 3a.

19. Find the length of the logarithmic curve y = ca*.

20. Find the length of an arch of the epicycloid,

X =

(o+

6)cos^ - 6

cos^^-i-^,

h

y = (o + 6)sin^ - 6sin^^-"-^^.b

21. Find the length of an arc of the evolute of the parabola y^ = 4 px,

namely,

27py3 = 4(x-2p)"

from the

pointwhere x = 2p to the

pointwhere x = 3p. Also, find the

length of arc of the preceding curve from the cusp (x = 2/)) to the point

where it intersects the parabola (atthe point for which x = 8j)).




32. Find the length of the arc of the curve y = logsinx between a; =

3

and 05 = -"

2

88. Find the length of the arc of the evolute of the circle,

a5 =

o(co8^ + ^sin^),

y =

o(8in^" ^co8^),

from ^ = 0 to ^ = a.

a

94. Find the entire length of the curve r = asin^-.

26. Find the length of arc of the spiral r = m^, from ^ = 0 to ^ = î.



CHAPTER X

APPLICATIONS TO MECHANICS

78. Mass and density. This chapter is introduced for the

purpose of giving the student further examples of the applica-ion

of the fundamental principle of the integral calculus, and

of affording him additional opportunity for practice in integra-ion.

The definitions of mechanics that are required in what

follows are merely stated, but are not discussed. They will be

familiar to thosewho

have had theadvantage of

an

elementarycourse in that subject.

Other readers can only assume these

definitions as data for problems in integration.

Ma^s, The mass of a body is usually defined as" the quantity

of matter which it contains," and is specified in terms of the

mass of a standard body. In English-speaking countries, for

ordinary purposes, the

standard

mass is a certain bar ofplati-um

marked "P.S. 1844. lib.," which is called the "imperial

standard pound avoirdupois," and is preserved at the Office of

the Exchequer in London. Any mass equal to this standard

mass is then a unit of mass. For scientific purposes in general,

and in countries where the metric system is adopted, the standard

of mass is the "kilogramme des archives," a bar of platinum

kept in the Palais des Archives in Paris. A mass equal to one

thousandth of this standard is then the unit of mass ; this unit

is called the gram. The mass of a body should not be con-founded

with its weight. The weight of a body depends upon

its distance from the center of the earth, but its mass is inde-endent

of its position.

Density. The mean density of a body is the quotient of its

mass by its volume. The density at a given point of a body is

167





78-79.] APPLICATIONS TO MECHANICS 169

smaller the portions Am the more nearly do they come to being

particles with distances rj, r^ """, from the fixed plane. Ulti-ately,

therefore, the distance of their center of mass from the

plane is given by

In accordance with the first definition of integration this is

written,

I rdm

Jdm

Therefore, in the case of any continuous distribution of matter,

the coordinates x, y, z, of the center of mass are given by

ixdm \ydm \zdtn(4)

\ dm \ dm \ dm,

Ifp

be the density at any point of a body, and dv an infini-esimal

volume about the point,

dm^^pdVf

the total mass =

Jp dv^

and formulae (4)become

_

\^dv_

\pydv_

iftzdv

^ = ^ ; y--^ ; z = ^ " (5)

\^dv \^dv \^dv

The densityp usually varies

frompoint

topoint of

a body,and

it is generally expressed as some function of the position of the

point. If the body is homogeneous,p

is constant and can be

removed from formulae (5)by cancellation. If jobe the mean

density of a non-homogeneous body, then, by the definition in

Art. 78,

rpcfv

P=S (6)



170 INTEGRAL CALCULUS [Ch. X.

If matter be supposed to be continuously distributed along a

line or curve making, as it were, a wire of infinitesimal cross-

section, or so thinly laid upon a surface, curvilinear or plane,

that the thickness of the layer may be neglected, the term "mass-

center"

can also be used with reference to lines and curves, sur-faces

and plane areas. If As, A^S',A^ are small elements of a

line or curve, a curvilinear surface, and a plane area respectively,

and pis the linear density or the surface density, the coordinates

of the mass-center of an arc, surface, or plane area are obtained

from formulae (5)on the substitution of ds, dS, dA respectively

for dV. Expressions for these differentials have already been

obtained in the preceding articles. The mean linear and surface

density can be obtained by making these substitutions in for-ula

(6).

Ex. 1. Find the total mass and the mean density of a very thin plate

which is the firstquadrant of the circle whose equation is x^"\-1/^

a^, and

whose density varies at each point as xy.

If p denote the density, then by the given

condition,

pccxy;

that is, p = kxyj

in which k is some constant.

If M denote the total mass of the quadrant,

and dm denote the mass of an infinitesimal rec-tangle

about any point,FiQ. 48.

M=^(dm=(pdA

= r('^'''"^kxydxdyJo Jo

= ika*.

If p be the mean density, it follows from the definition that,

(pdA

Jira2~2ir



79.] APPLICATIONS TO MECHANICS 171

Here,

Ex. 8. Find the center of mass of the thin plate described in Ex. 1.

ipxdv ikxyxdA

(pdv M

M

Similariy, y = A ""

Ex. 8. Find the mass-center for a thin hemispherical shell,radius a, whose

density at each point of the surface varies as the distance y from the plane of

the rim.

Let the hemisphere be described by revolving tlie semicircle of radius a

and center O about the y-axis O F, which is at right angles to the diameter,

the point 0 being takenfor

the origin

of coordinates. Let P, whose coordi-ates

are ", y, be any point on the

semicircle, and draw PJtf, PN at

right angles to the axes of x and y

respectively. Join OP^ and denote

the angle NOP by e.

At the point P, y = a cos ^ ;

also at P, p X y,

tliatis, p" ka cos 0,

in which k is some constant. The infinitesimal arc of length ds at P describes

a zone about OF whose area is given by

d8 = 2ifNPd8.

or, since ds = add, = 2 ira sin ^ . o dtf.

The symmetry of the figure shows that

x = 0.

rf

Also,

(pydS 2nka*

f'^^

cos^ 0 sin 0 d0

y =l ^

XpdS2vka^

i^ cos 08iR0d0

Hence, the center of mass is at the point (0,| a).




Hence,

Ex. 4. Find the center of mass of a lig^tcircular cone of height A, which

is generated by the revolution of the

line y =: ax about thex-axis, when

the density of each infinitely thin

croBB-section varies as its distance

from the vertex.

Symmetry shows that the center

of mass is in the z-axis. Suppose

that a very thin plate B8 is taken

which cuts the axis of the cone at

right angles at C at a distance x

from the vertex.

The radius CB of the cross-section = ax.

The density of this thin plate, p = kx.

The volume of the thin plate, dV=ir cS^ dx

Ex. 5. Find the mean density of the cone described in Ex. 4.

Ex. 6. Find the mass-center of the surface of the cone in Ex. 4.

Ex. 7. Find the mass-center of the cone generated in Ex. 4, and the mass-

center of its convex surface when the density is uniform.

Ex. 8. Find the mass-center of a quadrantal arc of the hypocycloid

asf-f y" = a'.

Ex. 9. Find the mass-center of the convex surface of a hemisphere of

radius 10.

Ex. 10. The quadrai\t of a circle of radius a revolves about the tangent

at one extremity ; prove that the distance of the mass-center of the generated

curved surface from the vertex is.876

a.

Ex. 11. Find the mass-center of the semicircle of x* + y^ = a2 on the

right of the y-axis.

Ex. 18. Find the mass, the mean density, and the mass-center of the

semicircle in Ex. 11 when the density varies as the distance from the

diameter.



79-80.] APPLICATIONS TO MECHANICS 173

Ex. 18. Find the mass-center of a circular sector of angle 2 a, taking

the origin at the center, and the x-axis along the bisector of the angle.

Ex. 14. Find the mass-center of the first quadrant of the ellipse b^^.

+ a2y2 = a^b^.

Ex. 16. Find the mass-center of the area between the parabola y^ = 4 az^

and : (a)the double ordinate for a; = ^ ; (b)the ordinate ioTx = h and ac-axis.

Ex. 16. Show that the mass-center of the circular spandril formed by a

quadrant of a circle of radius a and the tangents at its extremities is at a

distance

.2234

a from either tangent.

Ex. 17. Find the mass-center of a quadrant of the hypocycloid "" + y*

= a*.

Ex. 18. Find the mass-center of the area between the parabola x* + y*

= a^ and the axes.

Ex. 19. Find the mass-center of the area between the cissoid y^ =-^

"

and its asymptote.a

"

x

Ex. 20. Find the mass-center of the cardioid r = 2 a (1 " cos ^).

Ex. 81. Find the center of mass of the solid paraboloid generated by the

revolution of y^ = 4iax about the ic-axis.

Ex. 22. Show that the center of mass of a solid hemisphere of uniform

density and radius a, is at a distance | a from the plane of the base.

Ex. 28. Show that the center of mass of a solid hemisphere, radius a, in

which the density varies as the distance from the diametral plane is at a dis-ance

^ a from this plane. Also show that the mean density of this hemi-phere

is equal to the density at a distance f a from the base.

Ex. 24. Find the center of mass of a solid hemisphere, radius o, in which

the density varies as the distance from the center of the sphere.

Ex. 26. Find the center of mass of the solid generated by the revolution

of the cardioid r = 2 a (1 "

cos^Jaboutitsaxis.

80. Moment of inertia. Radius of gyration. If in any system

of particles the mass of each particle be multiplied by the square

of its distance from a given line; the sum of the products thus

obtained is called the moment of inertia of the system about that

line. Thus, if m^, m^, """, be the masses of the several particles,




**!" ""2" *"} ^êir distances from the line, and I denote the moment

of inertia,

/= miVi^-f

wigrs*-I-

"" ;

that is, /= l,m7^. (1)

In any case in which matter is continuously distributed, as in

a solid cylinder, a shell, etc., the matter may be supposed to be

divided into small portions, A??ii, Awij, """. By reasoning similar

to that employed in the last article,it can be shown that

If matter be supposed to be distributed uniformly along a line

or curve, or upon a curvilinear surface or a plane area, the term

"moment of inertia" can also be used in reference to curves,

surfaces, and plane areas.

Let M denote the total mass of a body, namely I dm, and I its

moment of inertia about a given line or axis. If k satisfies.the

equation

r1 r^dtn

that is,if A;2 = ^ =^

,

^

jam

k iscalled the radius of gyration of the

bodyabout the given

axis.

Ex. 1. Find the moment of inertia of a rectangle of uniform density,

whose sides have the lengths ft, d about a line which passes through the

center of the rectangle and is parallel to the sides of length ft.

The density per unit of area will be represented by unity. Let the axes

of X and y be taken parallel to the sides of the rectangle, the origin being at

the center, and let

AB^h, BC = d.



80.] APPLICATIONS TO MECHANICS 175

Then 1= (y^dA

d h

= j'j y^dydx

12*

This moment of inertia is impor-ant

in calculations on beams. Since

the mass of the i*ectangle = "d,

M 12*

*" = ^ =

^

dx

Udy

Fig. 51.

Ex. 8. Find the moment of inertia of a very thin circular plate of uniform

density of radius a about an axis through its center and perpendicular to

its plane.

Taking the density as unity per unit of area,

"Jo Jo r'^^rdrde

Also,

2

*

TO*

lc =L

=l^

=^.

M Tcd^ 2

Ex. 8. Find the moment of inertia about itsaxis of a right circular cone

of height h and base of radius 5, the density being uniform, and m being

the mass per unit of volume.

The moment of inertia is equal

to the sum of the moments of inertia

of very thin transverse plates like

B8, If

OC:=X,

then, by similar triangles.

Fig. 62.




Hence, if dl denote the moment of inertia of the plate B8 of thickness

(to,by Ex. 2,

Therefore, for the whole cone.

^X"""-'"to

10

Also, i^^L^J-îLBÎ^M mV \mrcl^h

Ex. 4. Find the radius of gyration of a uniform circular wire about its

diameter.

Ex. 6. Find the moment of inertia of the triangle formed by the axes and

a line whose intercepts are a and 6, about an axis which passes through the

origin, and is at right angles to the plane of the triangle.

Ex. 6. Find the radius of gyration about its line of symmetry of an

isosceles triangle of base 2 a and altitude h.

Ex. 7. Find the moment of inertia about the x-axis of the area between

the line and the parabola which both pass, through the origin and the point

(a,6),the axis of the parabola being along the x-axis.

Ex. 8. Find the moments of inertia of the ellipseh'h? + aV = a^'^ : (a)

about the x-axis ; (6)about the ^-axis ; (c)about an axis that passes.through

the center of the ellipse and is perpendicular to the plane of the ellipse.

Apply the results to the circle x^ + y^ = a^.

Ex. 9. Find the moment of inertia of the thin plate in Ex. 1, Art. 79,

about the x-axis.

Ex. 10. Find the moment of inertia of a homogeneous ellipsoid

aboutthe

x-axis.

Ex. 11. Find the moment of inertia of the surface of a sphere of radius a

about a diameter, m being the mass per unit of surface.

Ex. 12. Find the moment of inertia of a solid homogeneous sphere of

radius a about a diameter, m being the mass per unit of volume.

Ex. 18. Find the moment of inertia of the semicircular plate described in

Ex. 12, Art. 79, about the diameter.

Ex. 14. Find the moment of inertia, and the radius of gyration about its

axis, of a homogeneous right circular cylinder of length I and radius JB, m

being the mass per unit of volume. Also about a diameter of one end;



CHAPTER XI

APPROXIMATE INTEGRATION. INTEGRATION BY

MEANS OF SERIES. INTEGRATION BY MEANS

OF THE MEASUREMENT OF AREAS

81. Approximate integration. It was remarked in Arts. 4, 8

that in most cases in which a differential f(x)'dxis given it is

not possible to find the anti-differential. In some of these

cases, however, an expression can be found that will approxi-ately

represent the indefinite integral

jf(x)x. Even if this

cannot be done, it is often possible to determine a value that

will very nearly be that of the definite integral jf(x)dx.

Art. 82 explains a method, that of integration in series, by

means of which an indefinite integral may be expressed as a

function of x in the form of a series that contains an infinite

number of terms. An important application of this method

to another problem is given in Art. 83. Arts. 84--87 set forth

a method, that of measurement of areas, which reduces the

evaluation of a definite integral to a mere matter of careful

computation. In this connection several formulae for the ap-roxima

determination of areas are necessarily considered.

82. Integration in series. When the indefinite integral of a

given functiouj f(x)dx,cannot be found by any of the means

thus far considered, one of the most usual and most fruitful

methods employed is the following: The function f(x) is de-eloped

in a series in ascending or descending powers of x. If

this series is convergent within certain limits for x, the series

obtained by integrating it term by term is also convergent

177



178 INTEGRAL CALCULUS [Ch. XI.

within the.same

limits.* The greater the number of terms

taken the more nearly will the new series representif(x)dx,

Ex. 1. ""^'**'' *1. Find f-

(1 + a*)*

By the binomial theorem,

(l+ x^)*1 3^1.2 32 1.2.3 3"^

"

The second member is convergent for valaes of x between + 1 and " 1.

Integration of both members of (1)gives

^^^,2

_"L4.?_L^

g^^ 2.6.8 g^

(1+ ic^)*1

'

3 . 6 1 . 2

*

32 . 11 1 . 2 . 3'

3" . 16(2) J

The second member represents the required integral for values of x

between 4- 1 and " 1. It foUowafrom (2)that

/:dx

^

..2 2.6 1 2.6.8 1

(l+x*)*3.6'^1.2'38.11 1.2.3'3".16'^

Ex. 2. Find (e=^dx.

Since e" = 1 + " +1.21.2.3

(1) e"'=l + .^ +

j^+ ^^+.

which is convergent for all finitevalues of x.

* Suppose that

f(x) = ao + aix + 02*2 + ... + an-ix"-i + a^x" + .... (1)

Then Cf(x)dxa^ +

^+^+^..+^î^=^

+ ^^^^^ (2)^ 23 "n+l

The series in (1)is convergent when-^^

is less than unity for all values

of n beyond some finite number. The series in (2) is convergent when

" - "

.î^ând therefore when

^^, is less than unity for all values of n

n + 1 a"-i a"_i

beyond a certain number. Since theeonvergency of

bothseries

depends

upon the same condition, the second series is convergent when the firstis

convergent.



82-83.] APPROXIMATE INTEGRATION 179

Integration of both members of (1)gives

2.0 1.2.3.71.2.3.4.9

Ex.

(2) j'e''dxc +

a;+^+-The second member represents the required integral for values of x

between 4- 1 and " 1. It follows from (2)that

J-i V 3^1.2.6

1.2.3.7

1.2 .3.4.9 /

Ex 4 f ^ ^^- *" f"==" (Putsina;=;".)

^ ^

Ex. 9. f?!^.Ex.6. \x^\/\-v:^dx, "^

*

"Êx.10. J^ to.

Ex. 6. fa;2Vl-iC'^dx.

"^ Ex.11. f?!I^(to.(Compare Ex. 28, page 98.) J x

83. Expansion of functions by means of integration in series. A

function can be developed in series by means of the method

described in the last article if the expansion of its derivative is

known. The series which represents the function is obtained by

integrating the series which represents the derivative, and deter-ining

the value of the constant of integration.

Ex. 1. Expand tan"i a: in a series of ascending powers of x.

Differentiation and division give

d. tan-isc =

-^= (1 - a;2 + "" - jc" + ... + (- l)'Hc2* )dx,

I + a:*

which is convergent when x liesbetween " 1 and +1.

Integi'ating,

tan-ix = c +

x-^+^-?I+...

+ (-l)n"*"^'

--

3 6 7^ '

2n+l

The substitution of 0 for x gives

mv = c,

m being an integer ; and hence,

tan-la = mT +a;-^H-^-?^+

...

3 6 7

INTEGRAL CALC. " 13




This series* can be employed for values of x between " 1 and + 1. It

can be used for computing the value of v. For, on putting x = " therein,

it is found that "^

tan-" J-=m"- + 5:

= "",r +J- f1 -

i+i

-

-I-+ ... V

y/S" VS\ 0 46 189 /'

whence, , = 2v/3 (ll +"- J5+

...).Ex. 2. Expand t sin-^ a; in a series in x ; and compute the value of v by

putting x = J.

Ex.8. Derivet

j'e-.'dxl-^g ^^-j-^+....hich ta

convergent for all finitevalues of x.

Ex.4. Show that

log(a + x)=loga + ^-^,3^-^^-f...when|

and that log (a + X) = log a + ^-

-^+ ^ -

-^+ . " . when |a; |" 1.

x 2x^ oac* 4x*

The symbol |x \denotes the absolute value of x.

Ex. 5. Derive series for log (1 4- x),log (1 -

x),log 2, log 9.

Ex. 6. Develop log (x + Vl + x^)in a series by integrating (1+ x*)"^dx.

Ex. 7. " Show that

k^ being less than unity. (SeeEx. 9, Art. 46.)

* It is usually called Gregory's series, after its discoverer, James Gregory

(1638-1676). It was found also by Leibniz (1646-1716).

t This expansion is due to Newton (1642-1727),nd, by means of it,he

computed the value of ir.

t This integral is often met in the theory of probabilities, and in certain

questions in physics. For the evaluation of t "-**c?x when x is greater than

unity, see Laurent, Cours d*Analyse, t. III., " IV., p. 284. For the deriva-

tion of (e-'^dx = J Vir,see Williamson, Integral Calculus, Ex. 4, Art. 116.

" This integral is called the **

elliptic integral of the first kind." It re-ceived

the name ellipticintegral from its similarity to the integral in Ex. 8,

which represents the length of a quadrantal arc of an ellipse, and is known

as "the

ellipticintegral

ofthe

second

kind." The integralof

the first

andsecond kind are usually denoted by F(Jc,^),E(Jc,0), respectively. These

names and symbols were given by Legendre (1762-1833).




Ex. 8. Show that

(See Ex. 9, Art. 46 ; Ex. 7, Art. 71.)

Ex. 9. Deduce the value of t by means of the series in Ex. 1, itbeing

known that - = 4 tan- 1i tan- 1

-^.5 239

84. Evaluation of definite integrals by the measurement of areas.

It has been seen in Arts 4, 6, that the definite integral I f(x)dx,

may be graphically represented by the area included by the curve

whose equation isyz=f(x),the axis of x, and the ordinates for

whichx = ay x = b] and it has been

observed

that the

evaluation

of the integral is equivalent to the measurement of this area.

The numerical value of the integral j f(x)dx, which is also the

same as that of the area justdescribed, has been obtained up to

this point, by finding the anti-differentialof f(x)dXf say "t"(x),

substituting b and a for x therein, and calculating " (6) "^(a).

But when it is not possible to find the anti-differentialof f(x)dx,

recourse must be had to other methods.

While, on the one hand, as already shown, areas may be

determined by evaluating definite integrals, on the other hand,

definite integrals may be evaluated by measuring areas. If the

anti-differentialof f{x)dxis unknown, the value of j f(x)dxcan

be found in the following way. Plot the curve y=zf(x)from

x=:a to x = b, erect the ordinates for which x = a, a; = 6, and

measure the area bounded by the curve, the axis of x, and these

ordinates. There are several rules or formulae for determining

areas of this kind. The degree of approximation to absolute

correctness depends in general only on the patience of the

calculator. These formulae, some of which are usually given in

manuals for engineers, are called ^'formulae for the approximate



182 INTEGRAL CALCULUS [Ch.XI.

determination of areas," or "formulae for approximate quadra-ure."

They may be given the more general title, ^^formulce

for approximate integration.^^ The two rules most frequently

employed, namely, the trapezoidal rule and Simpson's one-third

rule, are discussed in Arts. 85, 86, and a rule deduced from them

is given in Art. 87. Other rules are given in the Appendix.*

It should be observed that only a numerical result is obtained

by means of these rules. The knowledge of the value of the

definite integral | f{x)dx thus calculated does not give any clue

whatever to the expression of the indefinite integral I f{x)dx as

a function of a?. If the indefinite integral \f{x)dx has been

found in the form of a series which is convergent for values of x

between a and 6, the value of the definite integral I f(x)dXy

can be found as accurately as one pleases by taking a suffi-iently

large number of terms. Illustrations of this remark

have been given in Exs. 1, 2, Art. 82, and in Exs. 1, 2, 7, 8,

Art. 83.

85. The trapezoidal rule. Let AK^ be a portion of a curve whose

equation may or may not be known ; and let LA^ TK, be drawn

at right angles to the line

OX, It is required to find

the area AKTL contained

between the curve AK, the

line XT, and the perpendicu-ars

LA, KT,

Divide LT into n parts,

each equal to h, and at the

points of division erect the

perpendiculars MB, NC, """,

8H, Draw the chords AB,

BO, """, HK. A

rule

for finding the area

of

LAKT

will

now be

" See Note E.





184 INTEGRAL CALCULUS [Ch.XI.

Ex. 2. Show that the approximate value obtained for'the above integral,

by making 20 equal intervals, is 333}.

ru

Ex. 8. Show that the approximate value of I \ogioxdx, unit intervals

being taken, is 7.990231.

86i The parabolic or Simpson's *one-third rule. The parabolic

rule for approximating to the value of the area LAKT is derived

by substituting parabolic arcs through ABCy CDE, """, GHK, for

thearcs

of the givencurve

passing through these points, theaxes of the parabolas being vertical, and then summing the

areas of the parabolic sections, LABGN, NGDEP, ..., RGHKT,

Fio. 64.

which are thus formed. A parabolic arc, as CDE, will more

nearly coincide with the given curve through CDE, than will

the chords CD, DE. For the purposesof

thisrule,

n the num-

berof equal parts into which LT \^ divided must be even, since

a parabolic strip is substituted for each of the consecutive pairs

of trapezoidal strips ; for example, NCDEP for ND-^

DP.

The area of one of the parabolic strips, say NCDEP will first

be found. Through D draw CE^ parallel to the chord CE, and

produce NC, PE to meet C'E^ in C, E\

" Thomas Simpson (1710-1761).




The parabolic strip NCDEP=: trapezoid NCEP+ parabolic

segment CDE,

The parabolic segment CDE = two thirds of its circumscribing

parallelogram CCE'E,

Hence, the parabolic strip

NCDEP= NP^NC+PE) + "{QZ" - |(J\r(7+^ j]

^2h(:^N0-\-iQD-îPE)

=hNO

+ ^QD-\-PE).o

Application of the latter formula to each of the parabolic strips

in order beginning with the first on the left,and addition, gives,

approximately,

area LAKT = â + 4 + 2 + 4 + 2 + . -. + 2 + 4 + 1),8

in which merely the coefficients of the successive perpendiculars

LA, MB, ..., TK are written. As in the case of the trapezoidal

rule, the greater the number of equal parts into which iT is

divided, the more nearly equal will the area thus calculated be

to the true area.

If the equation of the curve AK isy=f(x), and OL = a,

OT=b, and XT is divided into n parts, each equal to"~ ^,

then

lengths of the successive ordinates, LA, MB, ."" TK, are /(a),

ffa+^-^=^\fa-h2 ^^^\ ..., f(b). Hence, on calling these

successive lengths, yo, yi, t/a """ yn"

rf(x)cb:^(yo-\-4.y,-\-2y,^4.y,-\-2y..




For the sake of computation, this may be put in the form,

.2b-avl

X|^/(x)"fx=|.^^[i(yoyn) + 2(yi + y8 + - + y"-i)

+ (ya + y4 + - +

yf"-2)]*/"lO

Ex. 1. Evaluate I ic* "to by this method, taking n = 10.

Here, yoi yu ^2* """, yio? are 0, 1, 81, 625,..., 10,000, respectively; and

hence, approximately,

rVda;=j"{J^Si^+2(l+81+625+2401+6661)(16+256+1296+4096)}= 20001f

The true value of the given integral is 20,000 ; thus the error is only IJin 20,000.

ru

Ex. 2. Show that the value of I logioxdx calculated by this rule for

n = 10, is 8.004704 (compare Ex. 3, Art. 85).

A comparison between these two rules is given in the following

quotation :t "The increase in accuracy (ofthe parabolic)ver

the trapezoidal rule is usually quite notable, unless the number

of ordinates become large, in which case they both approximate

more and more closely to the true value and to each other. In a

* If n be the number of equal intervals into which the range 6 " a is

divided, the outside limit of error that the parabolic formula for integration

can have, is

\ 2 ) 90n*

in which av is some value of x between a and 6, and f^ (x) denotes the

fourth derivative of /(as). The outside limit of error in the case of the

trapezoidal rule is

12 n2 -^^^^'

in which /"(x) denotes the second derivative of /("). If " is doubled, the

limit of error is reduced, therefore, to ^ and J of itsformer amount. (See

Boussinesq, Cours dÂnalyse, t. II. 1, " 262, and Markoff, Differenzen-

rechnung, " 14, pp. 57, 59.)

t This is from an

article, entitled,**New Rules for Approximate Integrar-

tion,'* in the Engineering News (N. Y.),January 18, 1894, by Professor W.

F. Durand of Cornell University.




series of trials made by the author upon a number of integrals of

various forms for the purpose of testing the relative accuracy of

these rules, it was found for cases in which the locus was of single

curvature only that the trapezoidal rule required about double the

number of sections for equal accuracy with the parabolic rule.

Where the locus involves several changes of curvature, as in

lumpy and irregular curves, and the number of sections is moder-te,

one rule is as likely to be right as the other, and both are

likely to be considerably in error. For a large number of sec-tions,

however, the parabolic rule will show its superiority .as

above."

87. Durand's rule. From a discussion *on the trapezoidal and

parabolic rules. Professor Durand has deduced another rule for

which" it seems not unfair to claim substantially the full prob-ble

accuracy of the parabolic rule, and practically the simplicity

in use of the trapezoidal rule." It is as follows, merely the co-effici

of the successive ordinates being written in order from

the left:

approximate area= ^[3%-|-|f

1 4-1 + """ +1-hl4-}f

+1^];

or,

approximately,

area = 7^[.4+ 1.1 + 1 + 1 f """ + 1-f-

1 + 1.1 +.4].

The number of intervals may be even or odd.

reo"

Ex. 1. Find the value of I sin 6 d$ with 10" intervals.Jo

The circular measure of 10" is.17453.

The rule gives for the approximate

value of the integral,

J sin ede =

.17453[.4(sin" + sin 60")+ 1.1 (sin10" + sin 60")

^

+ (sin20" H- sin 30"-h sin 40")]=

.5000076.

reap r -iflo"Since the exact value of I sin ^d^ is " cos ^

,or

.6,the difference

between the above approximate and the true values of this integral is not

more than one

part

in 66,666.

* In the article mentioned in the preceding footnote.




J'M)^ dx calculated by this rule with unit intervals gives0

a difference of one part in 3333.

r\%

Ex. 3. Show that \ logio x calculated by this rule vnth unit intervals is

8.004062. (Compare with Ex. 3, Art. 86, and Ex. 2, Art. 86.)

88. The planimeter. Attention has been drawn to the fact that

the value of a definite integral is also the value of a certain plane

area, and that, consequently, the measurement of the area is

equivalent to the evaluation of the integral. In Arts. ^6, 86, 87,

rules are given for approximately determining plane areas, and

other rules therefor are given in the Appendix.* These areas can

be measured exactly by instruments called mechanical integrators

or planimeters. A planimeter measures the area of any plane

figure by the passage of a tracer round about the perimeter of the

figure, the readings being given by a self-recording apparatus.

There are several kinds of planimeters, but they all have certain

fundamental properties in common. The first planimeter was

invented by the Bavarian engineer, J. M. Hermann, in 1814.

Amsler's polar planimeter, which was invented by Jacob Amsler

when a student at Konigsberg in 1854, is the most popular on

account of its simplicity and handiness in use. Thousands of

them have been made at his works in Schaffhausen.

The Amsler planimeter is shown in Fig. 55. It consists of two

bars, (a)the radius bar, and (p)the pole arm, jointedt the point

C The tracing point P, which now coincides with the point B

of thefigure ABDE, is

carried roundthe curve,

and the rollerm,

which partly rolls and partly slips, gives the area of the figure ;

and by means of the graduated dial h, and the vernier v in con-nection

with the roller m, the result is given correctly in four

figures. The sleeve H can be placed in different positions along

the pole-arm h, and fixed by a screw s so as to give readings in

differentrequired units.

A

weightat w is

placed uponthe bar to

* See Note E.




keep the needle point in its place, but in instruments by some

other makers Tis a pivot in a much larger weight, which rests

on the paper. The accuracy of the reading depends upon the

accuracy with which the tracing point follows the curve.*

Fig. 65.

Professor O. Henrici's Report on Planimeters (Report of the British

Association for the Advancement of Science, 1894, pp. 490-523) contains

a sketch of the history of planimeters, the geometrical theory of gener-ting

areas, descriptions of early planimeters, a discussion on Amsler's

planimeter, and a description of some recent planimeters. Professor H. S.

Hele Shaw's paper on Mechanical Integrators (Proceedingsf the Institution

of Civil Engineers, Vol. 82, 1885, pp. 75-143) gives an account of the theory

and the practical advantages of several varieties of planimeters. The descrip-ion

given above is from this paper. An explanation of the theory of

Amsler's planimeter is given by Mr. J. MacFarlane Gray in Carr's Synopsis

of Mathematics. There is a discussion on planimeters in Professor R. C.

Carpenter's Text-bookof Experimental Engineering^ pp. 24-49.

* For the fundamental theory of the planimeter, see Note F, Appendix.



CHAPTER XII

INTEGRAL CURVES

89.Introduction. A firstintegral

curve was defined in Art. 15.

The student is advised to review that article thoroughly before

proceeding further. In this chapter the subjectfintegral curves

will be studied more fully, and some of their applications to

mechanics will be pointed out. Differentiation under the sign of

integration is an important topic in the integral calculus. Only

a

very specialcase, however, is

necessary

in

what

follows : this

case is considered in Art. 90. Arts. 92, 93, 94, contain an

exposition of the simpler properties of integral curves and a few

examples of their usefulness. Their applications are of especial

value to the student of engineering. For the proper understand-ng

of several of them, a better acquaintance with the theorems

of mechanics is required than some readers of the calculus may

be presumed to have at this stage. Accordingly, a further expo-ition

of the service that may be rendered by these curves is

given in the Appendix for purposes of future reference. Articles

94, 95, discuss the practical plotting of integral curves.*

90. Special case of differentiation under the sign of integration.

A special case of differentiation under the sign of integration

* Arts. 91-96 and the related matter in the Appendix are taken with some

slight but no essential change, from an article entitled Integral Curves,

by Professor W. F. Durand, Principal of the Graduate School of Marine

Engineering and Naval Architecture, Cornell University. The article, which

appeared in the Sibley Journal ofEngineering, January, 1897, is practi-ally

all reproduced here. This chapter has also had the benefit of Professor

Darand's revision.

190





192 INTEGRAL CALCULUS [Ch.XII.

Therefore, letting A6 approach zero,

^=nj\h-xr-^f(x)d^ (2)

This result will be required in Art. 93.

91. Integral curves defined. Their analytical relations. A more

general definition of an integral curve than that given in Art. 15

willnow be introduced. In

what

follows, a

number ofcurves

will be spoken of together. In order to distinguish between

them, the system of ordinates, that is, the j/'s,for each of the

several curves will be denoted by a subscript number.

If. y=f(s^),

or, for the sake of distinction,

y" =/("') (1)

be the equation of a given curve, the curve whose equation is

y'=\i'y'^''(^"

is called z,firstintegral curve of thecurve

whose equation is (1).The latter is called the fundamental curve. Since I ydx \^ oi

Jo

the second dimension, and yi should be linear, the constant factor

-is introduced in (2),n which a is a linear quantity and has a

magnitude that will make equation (2)convenient for plotting.

It may be called a scale factor. In the definition in Art. 15 the

scale factor was unity.

From (2)on differentiation,

dx a

Hence, as x varies, the slopes of the firstintegral curve vary as

the ordinates of the fundamental ; and therefore the former can

be represented by the latter,and vice versa.



90-91.] INTEGRAL CURVES 193

The firstintegral curve (2)also has a firstintegral, the latter

has a firstintegral, and so on. These successive integral curves

are called the second, third, etc.,integral curves of the original or

fundamental curve. On using the constant linear quantities, h,

c, " " " w, as scale factors for the sake of plotting the curves con-venientl

and on distinguishing by different subscripts the ordi-

nates that belong to the various curves, the latter will have the

following equations:

Fundamental, y^ =/("); (1)

1 Cfirstintegral, yi "

-\y(i^] (2)

second integral, 2/2 T I yi^" =-r I I ôC^; (3)

oJo aoJi) Jo

1 /"" 1 /"x /*x r*x

third integral, 2/3= - I 2/2^^ =

^- ( ( \ y^^\ (4)c*/o aoc*/o

*/o "/o

nth integral curve,

yn = - f Vn-idx = " \ ) ) ... fyocbf". (5)wJo aoc"''wJo Jo Jo Jo

From equation (2)

and hence, ^=

^^^ (^)

And in general.

Equation (8)shows that as x varies, the wth derivatives of the

nth integral curve vary as the ordinates of the fundamental

curve ; and therefore, the former can be represented by the latter.



194 INTEGRAL CALCULUS [Ch. XII.

92. Simple geometrical relations of integral cnryes. In Fig. b^

RP, OA, OBj OCy represent the fundamental, and the first,

second, and third integral curves respectively, whose equations

are (1),(2),(3),(4),f Art. 91.

(a)As X increases, and so long as the fundamental curve RP

lies above the oj-axis,the ordinates of the first integral OA will

increase,

andthe tangent to OA

will makea

positive angle with

"

the aj-axis; when RP lies below the avaxis the tangent to OA

makes a negative angle with the oj-axis; when RP crosses the

a?-axis,the tangent to OA is parallel to the ar-axis. These prop-rties

follow from equations (2)and (6),rt. 91.

(b)At points for which the ordinate of the firstintegral curve

is a maximum or a minimum,-

and there also, by (6),

dx'

yo = 0.

Hence, to a zero value of the ordinate of the fundamental there

corresponds a maximum or a minimum value of an ordinate of

the firstintegral curve.




(c)At points where an ordinate of the fundamental is a maxi-um

or a minimum,

and at points where the first integral curve has a point of in-lexion

Differentiation of the members of equation (6)shows that

g =

Owhenf=0.ar.

ax

Hence, to a point on the fundamental at which there is a

maximum or a minimum value of the ordinate, there corresponds

a

point of

inflexion on the firstintegral curve.

93. Simple mechanical relations and applications of integral

curves. Successive moments of an area about ailine. If each in-inites

portion of a plane area be multiplied by its distance

from a given line, the sum of all these products is called the

^momentoffirstegree of the area about the line. If each of the

infinitesimal portions of the area be multiplied by the square of

its distance from the given line, the sum of all the products is

called the moment of second degree of the area with respect to the

line. The latter is the moment of inertia of the area about

the line,examples of which were shown in Art. 80. The moment

of firstdegree is usually called the statical moment. In general,

if each infinitesimal portion of an area be multiplied by the nth

power of its distance from a given line, the sum of all these

products is called the moment of the nth degree of the area about

the line. For the sake of brevity, this may be called the nth

moment.

Thus (Fig.bQ),lay off OX = x^, and erect the ordinate AP at

X, and consider I ^y^dXj the area ORPX, between the funda-

INTEGRAL CALC. 14



196 INTEGRAL CALCULUS [Ch. Xll.

mental, the axes, and the ordinate AP, If the successive moments

of this area be taken about the ordinate AP for which x = x^ and

these moments be denoted by M^ M^t """, Jlf",in order, then

the firstmoment, î = | ^(xi-'X)ydx\ (1)

\ {xi-'X)h/dx; (2)0

the nth moment,-^n

= j *(î^Tv

^* (3)

In this notation, Jô = ) *(*i^Tv4^f

= j V ^^9 ^^^ area. (4)Jo

(a)Differentiation of (3)with respect to Xi will give by equa-ion

(2),rt. 90,

^=nr\x,-xy-'yd^;dxi Jo

that is, ^=nM^_i. (5)dxi

Hence, M^^^nC^

M^_idxi.

Since dxi is an infinitesimal distance along the aj-axis,it can be

written dx, and hence

M, =

n"'M,_^d^.(6)

By successive application of (6)there will finallybe obtained,

M^^nl P f'-r^Modaf^. (7)Jo Jo Jo

That is,the nth moment of the area ORPX about an ordinate

distant Xi from the origin is equal to factorial n times the nth



93.] INTEGRAL CURVES 197

integral of the moment of degree zero for the same area. On

substituting for Mq in (7)its value from (4),here is obtained,

Hence, the value of the wth moment of the area of y =/(")above described about an ordinate distant Xi from the origin, is

equal to factorial n times the ordinate of the (n + l)th integral

curve

atx =

Xi] and, therefore, the nth moment may be repre-entedby this ordinate. On using T^î to denote the ordinate of

the (wH- l)thintegral curve at a? = x^, this may be expressed by

In particular, the statical moment (1)is represented by the

corresponding ordinate of

the

second

integral curve,

andthe mo^

ment of inertia (2)by twice the corresponding ordinate of the

third integral curve. Thus in Fig. 56,

the area ORPX is represented by AX\

its statical moment about AP is represented by BX\

and its moment of inertia about AP by 2 CX,

Suppose that the scale factors used in plotting the three inte-ral

curves, each from the one of next lower order, are a, 6, c,

respectively, as indicated in equations (2),(3),(4),Art. 91.

Then,

zxQ2iORPX=a'AX',

thestatical

momentof

ORPXabout

AP =

ah

" BX]

the moment of inertia of ORPX about AP= 2 abc " CX:

(b) If G is the center of mass (orcenter of gravity)fORPX,

its distance HX from AP, by Art. 79, is determined thus :



198 INTEGRAL CALCULUS [Ch.XII.

(c) If k is the radius of gyration of the area OBPX about

AP, then by (Art.80),

, o_

Moment of Inertia about AP_

2 abcy^" 2 ftc

^^

Area ayi AX

For further applications to mechanics, and some general re-marks

on the use of these curves in engineering problems, see

Appendix. The reader is recommended to glance at the latter

remarksnow.

94. Practical determination of an integral curve from its funda-ental

curve. The integraph. Suppose that the equation of the

fundamental curve is y =/(").The ordinates of the firstintegral

curve that correspond to successive values x^^ x^ '"f,^n9 of the

abscissas are

- fydx,fldx,.., -r

ydx,

respectively. These may be determined by the ordinary rules

for integration when the functions under the sign of integration

are integrable. If the latter condition does not hold, recourse

can be had to some of the various methods of mechanical and

approximate integration described in Arts. 85-88. It will be

necessary to do this also, when the fundamental has been plotted

merely from a knowledge of the ordinates that correspond to

particular abscissas, the equation of the curve being unknown.

For example, in Fig. 57, the area of each successive section

between the ordinates of the fundamental may be found with a

planimeter, and the ordinates of the integral curve, which is

shown by the dotted line, may be found by successive additions.

As an instrumental check, it is well from time to time to go

around the entire area between the t/-axisand the ordinate in

question, and compare the result with the total area summed to

that point. Numerical means of integration may also be em-ployed.

The trapezoidal rule and the parabolic rule can be

readily used for finding successive increments of area in the case




of the fundamental, and hence for finding successive increments

of the ordinates of the firstintegral curve.

In whatever way the integral curve may be derived from the

fundamental, it is well, after plotting, to compare the two and

note the fulfillment of the simple geometrical relations, (a),(b),

Fm, 67.

(c),f Art. 92. Thus, one should look for a maximum or a mini-um

ordinate in the integral corresponding to every zero ordinate

in the fundamental, and for a point of inflexion in the integral

for each maximum or minimum in the fundamental. The tan-ent

of the integral varies with the ordinate of the fundamental,

and hence, the slope of the integral should increase or decrease

when the ordinate of the fundamental increases or decreases.

These relations may be noted in the curves in "Figs. b%j 57.

The integraph is an instrument that is used for drawing the

first integral curve from its fundamental. The theory of it is

given in the Appendix (Note G). It may be used also for

determining the area between a curve and the a?-axis. For the



200 INTEGRAL CALCULUS [Ch. XII

integral curve can be drawn with the integraph, and the ordinate

corresponding to the area can be measured. Since the length

of the ordinate represents the area, the latter can be found

immediately on making allowance for the scale-factor.

95. The determination of scales. In order to have the various

curves convenient for plotting, it is usually necessary to employ

different scales for the ordinates. If numerical integration is

used,, the value of the area of the fundamental will be found

directly, and the scale may be correspondingly selected so that

the curve will be kept within the desired limits as to size. If

the planimeter is used, the result will be given in square inches

or other area units, and must be converted into the value desired

by the use of a scale factor. Suppose the fundamental plotted

as follows:

horizontally 1 unit of length = p units of abscissa,

vertically 1 unit of "^length = q units of ordinates.

Then 1 unit of area on the diagram will represent pq units of

the integrated function, and the area found must be multiplied

bythis

factor inorder

toreduce

it to thevalue of

the integral

desired. The scale factors a, ", c, etc., may then be chosen as

before.

* See Note G.



CHAPTER XIII

ORDINARY DIFFERENTIAL EQUATIONS

96. Differential equation, order, degree. A few differential

equations which frequently appear in practical work will now

be discussed very briefly.*

A differentialequation is an equation that involves differentials

or differential coefficients. Ordinary differential equations are

those which contain only one independent variable. For example,

dy = cosxdx, (1)

,

HWV

da?*

y =

x^+ ^, (4)

are ordinary differential equations.

The order of a differential equation is the order of the highest

derivative that appears in it. The degree of a differentialequa-ion

is the degree of the highest derivative when the equation is

* For fuller explanations tjian are given here, reference may be made to

Introductory Course in Differentialquations, by D. A. Murray. (Long-ans,Green, " Co.)

201



202 INTEGRAL CALCULUS [Ch. XIII.

free from radicals and fractions. Of the examples above, (1)is

of the firstorder and firstdegree, (2)is of the second order and

firstdegree, (3)is of the second order and second degree, (4)is

of the firstorder and second degree, (5)is of the first order and

firstdegree. Differential equations of a very simple kind have

already been considered.

97. Constants of integration. General and particular solutions.

Derivation of a differential equation. If a

relationbetween the

variables together with the derivatives obtained therefrom satis-ies

a differential equation, the relation is called a solution or

integral of the differential equation. For example,

y = m sin x, (1)

y = n cos a?, (2)

y =' Acqs X-i-

B sinx, (3)

y = c sin (a; a), (4)

in which m, w. A, B, c, a, are arbitrary constants, are all solutions

of the equation

This may be verified by substitution. It will be observed that

(5)does not contain vi, n, A, B, c, or a. The solutions of the

differential equations of the first order, which have appeared in

the former part of this book contain one constant of integration ;

those of the second order contain two constants. Examples have "

been given in Arts. 8, 9, 12, 59, etc.

Solutions (1)and (2)above contain one arbitrary constant, and

solutions (3) and (4) each contain two. The question is sug-ested

: How many arbitrary constants should the most general

solution of a differential equation contain ? The answer can in

part be inferred from a consideration of one of the ways in which

a differential equation may arise, namely, by the elimination of

constants, On comparing (3)and (5)it is seen that (5)must



96-97.] ORDINARY DIFFERENTIAL EQUATIONS 203

have been derived from (3)by the elimination of the two con-stants

A and B, In order to eliminate two quantities, three

equations are necessary. One of these is given, and the others

can be obtained by successive differentiation. Thus,

y =

-4cos X'{- B sin x,

-2= " J. sin a?

-f-J5 cos ",

dx

-4= --

-4cos X " BsiD.X'f

dor

whence, " ^+ y = 0.

In order to eliminate three constants- from a given equation,

four equations are required. Of these, one is given and the

remaining three can be obtained only by successive differentiar

tion. The third differentiation will introduce a differentialcoeffi-ient

of the third order, which accordingly will appear in the

differentialequation that is formed by the elimination of the three

constants. In general, if an integral relation contains n arbitrary

constants, these constants can be eliminated by means of w + 1

equations. The latter consist of the given equation and n rela-ions

obtained by n successive differentiations. The nth differ-ntiation

introduces a differential coefficient of the ntl?order,

which will accordingly appear in the differential equation that

arises on the elimination of the constants. The solution of an

equation of the nth order cannot contain more than n constants ; .

for if it had n

-f-1, their

elimination would

lead to theequation

of the n-f

1th order.*

The solution that contains a number of arbitrary constants

equal to the order of the equation is called the general solution or

the complete integral. Solutions obtained therefrom by giving

* For a proof that the general solution of a differential equation contains

exactly n arbitrary constants, see Introductory Course in Differentialqua-

tions, Art. 3 and Note C, Appendix.



204 INTEGRAL CALCULUS [Ch.XIII.

particular values to the constants are called particular solutions.

For example, (3)and (4)are general solutions of (5),nd

y = 2cosajH-3sina:, y = 5 cos a: " sin a:, y = 2sin(a; ir),

are particular solutions.

Ex. 1. Eliminate the arbitrary constants m and c from

(1) y = tiMc + c.

Differentiating twice, (2) ^= m,

dx

C3)g0.

These equations may be.interpreted geometrically. If w, c, are both

arbitrary, (1)is the equation of any straight line ; and, therefore, (3)is the

differentialequation of all straight

lines. Ifcis

arbitrary andm has a definite

value, (1)is the equation of any line that has the slope w, and, accordingly,

(2) is the differential equation of all the straight lines that have the slope m.

Ex. 8. Find the differential equation of all circles of radius r.

The equation of any circle of radius r is

in which a, b, the coordinates of the center, are arbitrary. The elimination

of a and h gives

{"-(l)"}'"S.the equation required.

Ex.8. If y = âj2 + j5^ prove thata;^-^0.

dx^ dx

Ex. 4. Eliminate c from

y

= cas + c " c*.

Ex. 6. Form the differentialequation of which c^ + 2 cxeff + C^ = 0 is the

complete integral.

Ex. 6. Eliminate the constants from y = ax-\- hx^.

Ex. 7. Form the differential equation which has y = a cos (mx + 6) for

its complete integral, a and h being arbitrary constants.




Section I. Equations op the First Order and the First

Degree.

98. Equations in which the variables are easily separable.

If an equation is in the form

its sohition^ obtainable at once by integration, is

Jf.ix)x

-hfAiy)dy = c.

Some equations can easily be put in this form.

Ex. 1. Solve (1 -

x)dy- (1+ y)dx

= 0.

This may be written,

-^ -^-

= 0.^

1 + y 1-a;

This step is called**separation of the variables."

Integrating, log (1 + y) + log (1 -

x) = Ci,

or, (1+ y)(l-

a;) e"i = c.

Ex.2. Solve ^+ JL=J^ 0.

dx ^l-x^

Ex.8. Solve 3"*tanyda5 + (l-c")sec2y(fy 0.

99. Equations homogeneous in a; and y. If an equation is homo-eneous

in X and y, the substitution

y=:vx

will give a differentialequation in v and x in which the variables

are easily separable.

Ex. 1. Solve (x*+ ^)dx -2xydy= 0,

Rearranging. (1)| = Ôn putting y = vx,

dx dx



206 INTEGRAL CALCULUS [Ch.XIII.

Sabstitution of these values in (1) gives

'

dv 1 + r*

dz 2v

Separating the variables,^

-

^^^= 0.

X 1 " r*

Integrating, log a;(l r^)log c ;

that is,

loga;(l-?!Wlogc;

whence, a^-

y*=

ex.

Ex. S. Solve y^dx + (xy + x^)dy = 0.

Ex. 8. Solvex^dx-{7fi -\-y^)dy

0.

Ex. 4. Show that the non-homogeneous equation of the firstdegree in x

Budy

dy_

ax-\-hy-\-c

dx a'x + b'y + c'

is made homogeneous, and therefore integrable, by the substitution

x = xi + h, y = yi + kj

hf k being solutions of ah -\-bk

-\-c = 0^

a'h + b'k + c' = 0.

100. Exact differentialequations. An exact differential equa-ion

is one that is formed by equating an exact differential to

zero. It follows from Art. 24 that

Mdx-h

Ndy = 0

is an exact differentialequation if

dy dx

Ex. 1. Solve (a2-2xy- y^)dx- (" + y^dy = 0.

Ex. 3. Solve (x^ ixy-2y^)

dx + (y^ 4xy-2x^)dy

= 0.

Ex. 8. Solve (2xhf + iofi-I2xy^

+ Sy^-a^ + e^^dy

+ (l2x^y-\-2xy^-{'43fi-4y^-^2ye^-eif)dx 0.




101. Equations made exact by means of integrating factors. The

differentialequation

is not exact. Multiplication by " gives

This is exact, and its solution is- = c, or a? = (^.y

When multiplied by " f the firstequation becomes

xy

X y*

which is also exact. The solution is

logo? " logy = log c,

whence,- = (^ or a? = cy.

Another factor that will make the given equation exact is

--. Any factor such as ","

," employed above, which changes

aj* f ix^y

an equation into an exact differential equation, is called an

integrating factor. It can be shown that the number of integiât-

ing factors is infinite. There are several rules for finding

integrating factors. In the following examples, the necessary

integrating factorcan be found by inspection.

Ex. 1. Solve y dx " xdy -\-\ogxdx = 0.

Here, logxdx is integrable, but a factor is needed for ydx " xdy.

Obviously " is the factor to be employed, as it will not affect the third term

x^1

injuriouslyrom the point of view of integration. On multiplication by "

the given equation becomes

ydx-xdy . loga;_Q

x2"

as*

*



208 INTEGRAL CALCULUS [Ch.Xlll.

The solution of this equation reduces to

cx + y " log* " 1 = 0.

Ex. 2. a (xdy + 2 y dx) = xydy.

Ex.8. (x2+ y2+ l)dJB-2a;ydy = 0.

Ex.4. (x"c*-2wy2)dx + 2ma;ydy = 0.

102. Linear equations. If the dependent variable and its de-ivati

appear only in the firstdegree in a differential equation,

the latter is said to be linear. The form of the linear equation

of the firstorder is

|+iV=Q, (1)

in which P and Q are functions of x, or constants. The linear

equation occurs very frequently. The solution of

dx

that is,of^

=

^PdXyy

- \pdat \pd*IS y=:ce ^

,OT ye* =c.

On differentiation the latter form gives

J'^(dy-^Pyd^)0,

which shows that e''^ is an integrating factor of (1).

Multiplication of (1)by this factor gives

J'îdy"{-Pyda)

==J'^Qdx',

and this, on integration, gives

yeJ^^ = jgJ^'Q da?

-I-c,

or y =

e-^'''^{fJ'''^Qdx-^c}.2)




The latter can be used as a formula for obtaining the value

of 3/ in a linear equation of the form (1).

Ex. 1. Solvejc^-y =:sc8.

dx

This is linear since y and -âppear only in the firstdegree. On putting

dx

it in the ordinary form (1),it becomes

ax X

1 \pd* " r*^ ^1

Here P = "

, andhence, the integrating factor e = "

*= e'"'" = -.

X X

By using this factor, and adopting the differential form, the equation is

changed into

-dy"

-ydx= xdx,

X x^

Integrating, 2f= ^^ + c, or y =

-x* + ex.

"B2 2

Ex.2. Solve !^+y=:e-".dx

Ex. 8. Solve ^+ ^"^y = 1.

dx "2

Ex.4. Solve

^4--^y=

dx x^ + r (a;a+l)8

Ex.6. Solve ^+

iiy="

dx X xn

103. Equations reducible to the linear form. Sometimes equa-ions

that are not linear can be reduced to the linear form. One

type of such equations is

in which P, Q, are functions of x, and n is any constant. Divi-ion

by y" and multiplicationby (" n H- 1)gives

(- n +l)2/-"g+

- n + l)Pr"î = (- n-f-1)Q. .



210 INTEGRAL CALCULUS [Ch. XllL

On substituting v for ^"^^ this reduces to

^ + (l-n)Pv={l-n)Q,

which is linear in v.

Ex.1. Solve ^+iy

= *y.dx X

Division by y* gives y*^ +-jr*

= a^.dx X

On putting v for y^^ this takes the linear form

dv 5^ft^2

V = " OX*.

dx X

The solution is v = y* = ex* + fo^.

Ex. S. Solve ^+?y

= 3a;*y*dx X

Ex.3. Solve ^+

-^L= xyi

Ex. 4. Solve 3^ +-?-y

=^

daj 05 + 1 y*

Ex. 6. Show that the equation

in which P, Q, are functions of x, becomes linear on the substitution of v

Section II. Equations op the First Order but not of

THE First Degree.

104. Equations that can be resolved into component equations of

the first degree. In what follows,-^

will be denoted by p. Thedx

type of the equation of the firstorder and nth degree is

which on expansion becomes

p- + P,p-' + P^"-2 + ... + P".ip + P^ = 0. (1)




Suppose that the first member of (1) can be resolved into

rational factors of the firstdegree so that (1)takes the form,

(i"-i?0(l"-^2)-(i"-i?")= 0. (2)

Equation (1)is satisfied by any values of y that will make a

factor of the first member of (2)equal to zero. Therefore, in

order to obtain the solutions of (1),equate each of the factors

in (2)to zero, and find the integrals of the n equations thus

formed. Suppose that the solutions derived from (2)are

fi(x,y,(h)0, /2(aJ,, C2)=0, ..., /"(a?,2/,c")=0,

in which Cj, Cg, """, c^ are arbitrary constants of integration. These

solutions are justas general if Cj = C2 = """ = c^ since each of the

c's can take any one of an infinite number of values. The solu-ions

will then be written

/i(a?,f c)= 0, /2(a?,, c)= 0, ..., /"(a?,, c) 0,

or simply, /i(a?,, c)fâ?,, c)"

"/"(a?,, c)

0.

Ex. 1. Solve f V+ (X 4- y)^ + "y = 0.

\dxl ax

This equation can be written (P + y)(l"+ x)= 0.

The component equations are l" + y = 0, p + x = 0,

of which the solutions are log y + " + c = 0, 2 y + x* + 2 c = 0.

The combined solution is (logy 4- " + c)(2y 4- a;^ + 2c)= 0.

Ex. 3. Solve (^ ) = ax*. Ex. 8. Solve p8 + 2 xp2 _ ^2^2 _ 2xy^ = 0.

105. Equations solvable for y. When equation (1),Art. 104,

cannot be resolved into component equations, it may be solvable

for y. In this case,

can be put in the form y = F(x,p).

Differentiation with respect to x gives

P

nTTBGRAL CALC. " 15

=

*('^'^'i)




which is an equation in two variables x and p. From this it may

be possible to deduce a relation

^(a?,p,)=0.

The elimination of p between the latter and the original equa-ion

gives a relation that involves a?, y, c. This is the solution

required. If the elimination of p is not easily practicable, the

values of x and y in terms of 2" as a parameter can be found, and

these together will constitute the solution.

Ex. 1. Solve x " ffp = ap'^.

Here y =^"

P

Differentiating with respect to x, and clearing of fractions,

(ap"+

x)J=D(l-l)").Tliis can be pat in the linear form

dx 1_

ap

Solving, X =^

{c+ a sin-^p).

Substituting in the value of y above,

y = -

ap-\-"=(6 + a sin-ip),

Ex. 3. Solve 4y = jB"+ pa.

Ex. 8. Solve y = 2p + 3p2.

106. Equations

solvable

for x. In this case

f(x,y, p)can be

put in the form

x=zF(rfyp).

Differentiation with respect to y gives

r"-l"from which a relation between p and y may sometimes be obtained,

say, f(yyP,c)=0,




Between this and the given equation p may be eliminated, or x

and y may be expressed in terms oip as in the last article.

Ex. 1. Solve x = y + a logp.

Ex.3. Solve x(l+p*)=l.

107. Clairaut's equation. Any differential equation of the first

order which is in the first degree in x and y comes under the

cases discussed in Arts. 105, 106. An important equation of this

kind is thatof

Clairaut.* It has the form

y=px+f{p). (1)

Differentiation with respect to x gives

From this.p^p+\x+np)\f^.

ar+/'(p)

0, (2)

1 = 0. (3)

From the latter equation it follows that p = c. Substitution of

this value in the given equation shows that

y =

cx+f(c) (4)

is the general solution. See Introductory course in differential

equations, Art. 28, for remark on equation (2). Some equations

are reducible to Clairaut's form, for instance, Ex. 2 below.

Solution (4)represents a family of straight lines. The en-velope

of this family of lines will also satisfy the differential

equation, since a;, y, p, at any point on the envelope is identical

with the ", y, p of some point on one of the tangent lines of which

(4)is the equation. The equation of the envelope of (4)is called

the singular solution of (1). Singular solutions are discussed in

Chapter IV. of the work referred to above.

* Alexis Claude Clairaut (1713-1766) was the first person who had the

idea of aiding the integration of differential equations by differentiating

them. He applied it to the equation that now bears his name, and published

the method in 1734.



214 INTEGRAL CALCULUS [Ch. Xlll.

Ex. 1. Solve y-px + aVl+jp^.

The general solation, obtained by the substitution of c for p, is

y=:cx + aVl + c^

which represents a family of lines. The envelope of this family is the circle

a;2 + ys = a^.

The latter equation is the singular solution.

Ex. 8. x\y-px)=t/jif^.

On putting x^ = u, y^ = ", the equation becomes

which is Clairaut^s form. The solution is

that is, y^ = ca^ + "^.

Ex. 8. y =pz + sin-ip.

Ex. 4. py =p2x + TO.

Ex. 6. xy^ =pyx^ + x H-py.

108. Geometrical applications. Orthogonal trajectories. curve

is often defined by some property whose expression takes the

form of a differential equation. In the examples given below the

differential equations of the curves are of a less simple character

than those which appeared in similar problems in Arts. 8, 12, 32.

Problems that relate to orthogonal trajectoriesre of consider-ble

importance. Suppose that there is a singly infinite system

of curves

/(a?,2^,a)=0, (1)

in which a is a variable parameter. The curves which cut all the

curves of the given system at right angles are called orthogonal

trajectoriesf the system. The elimination of a from (1)gives

an equation of the form

"(",|)=0, (.)



107-108.] ORDINAHY DIFFERENTIAL EQUATIONS 216

the differential equation of the given family of curves. If two

curves cut at right angles, and if 0i,"^2"the angles which the

tangents at the intersection make with the axis of Xy then

and therefore, tan 4"i= " cot "^

Hence,

-^

for one curve is the same as for the other.

dx dy

dx

Therefore, the differential equation of the system of orthogonal

trajectoriess obtained by substituting

" " f ^dy dx

in equation (2). This gives

4"(",-!)=".Integration will give the equation in the ordinary form.

Suppose that /(r, ,c)

= 0

is the equation of the given family in polar coordinates, and that

is the corresponding differential equation obtained by the eliminar

tion of the arbitrary constant c. Leti/î,p^

denote the angles

which the tangents to one of the original curves and a trajectory

at their point of intersection make with the radius vector to the

point. Then

tani/fj

" cot \ff^

dBNow tanî = r" . Hence the differential equation of the

dr

required family of trajectoriess obtained by substituting

r dO 'dr



216 lifTEGItAL CALCULUS [Ch.Xllt

that is,

-r^^for^dr dO

in (3). This gives 4/r,,-

r^jâs the differential equation of the orthogonal system.

Ex. 1. Find the orthogonal system of the family of parabolas

Differentiating,y^

= 2 a,

dx

and eliminating a, y = 2x^"dx

This is the differentialequation of the given family. Substitation of

dy dx

ndx

gives y =

-2a;" ,

the differential equation of the family of trajectories.ntegration gives

the equation of a family of ellipses whose foci are on the y-axis, and whose

centers are at the origin.

Ex. 2. Find the orthogonal trajectoriesf the cardioids

r = o(l"

COBB).

Differentiating, " = a sin ^.

dO

dv (t

Elimination of a gives -i.= r cot ^"

dB 2

the differential equation of the given family of curves. Therefore, the equa-ion

of the system of trajectoriess

-r^= cot?.

dr 2

Integration gives r = c (1 + cos ^),

another system of cardioids.



108.] ORDINARY DIFFERENTIAL EQUATIONS 217

Ex. 8. Find the curve in which the perpendicular upon a tangent from

the foot of the ordinate of the point of contact is constant and equal to a,

determining the constant of integration in such a manner that the curve shall

cut the axis of y at right angles.

Ex. 4. Find the curve whose tangents cut off intercepts from the axes the

sum of which is constant

Ex. 6. Find the curve in which the perpendicular from the origin upon

any tangent is of constant length a.

Ex. 6. Find the curve in which the perpendicular from the origin upon

the tangent is equal to the abscissa of the point of contact.

Ex. 7. Find the orthogonal trajectoriesf the straight lines y = cx,

Ex. 8. Find the curves orthogonal to the circles that touch the y-axis at

the origin.

Ex. 9. Find the orthogonal trajectoriesf the family of hyperbolas

zy = A;2.

Ex. 10. Find the orthogonal trajectoriesf the ellipses

in which X is arbitrary.

^t y

-1

Ex. 11. Show that the system of confocal and coaxial parabolas

y*= 4

a(x+

a)is

self-orthogonal.

Ex. 12. Find the orthogonal trajectoriesf the system of circles

y = c cos d,

which pass through the origin and have their centers on the initialline.

Ex. 18. Find the orthogonal trajectoriesf the system of curves

y^sin nd = a".

Ex. 14. Find the equation of the system of orthogonal trajectoriesf the

2 afamily of confocal and coaxial parabolas r = " "

1+ cos ^

Ex. 15. Determine the orthogonal trajectoriesf the system of curves

)"" = a" cos n9 ; and therefrom find the orthogonal trajectoriesf the series of

lemniscates r^ = a^ cos 2 $,



"3 ^fx -Tit g-rnnaiBif rf "ria-

"", 4 ^4y* Z/"*^^Pra-";=0, sobjeettoAe oonditkn that f = 0

110" Sqiuiti4MMof tlieform ^^ =/(y)*Multiplication of both

\\\m\\wiv%ot th\H equation by 2^^-ives(IX

rntfl"mtinK, (dxj^^f-^^^'^'^'^'^





220 INTEGRAL CALCULUS [Ch.XIU.

an equation of the firstorder between p and x. Its solutiongives

p in terms of x ; thus,

The value of y can be found from this by successive integration.

Ex. 1. Find the curve whose radius of curvature is constant and equal

to J?.

The expression of the given condition gives

{" (!)"}"

Substituting p for ^, clearing of fractions,and separating the variables,dx

dp dx

4 ^

Integrating, ^= "

x " a

in which a is an arbitrary constant of integration.

Solvingfori), t, = ^ = ^^-"*

^ Vi22-(x-a)"

whence, y-b = " Vija-(x- a)3,

in which b is the arbitrary constant of integration. This result may be

written

(a;-rt)2(y-6)2 = iP.

The integralrepresents allcirclesof radius B,

Ex.2. Solve ^-o (i)"="-

Ex.8. Solve ^^ = 2.

dx^ dx^

112. Equations of the second order with one variable absent.

(a)Equations of the form

i^-%')""- w

s

\

\



111-112.] ORDINAHY DIFFERENTIAL EQUATIONS 221

on the substitution of p for

-^,become

dx'

/(|,i",^)0, (2)

an equation of the firstorder in p and x. Suppose that the solu-ion

of (2)is

Then the solution of (1)is y = jF(xy Ci)a?+ c^

(6)Equations of the form

ff^, y\=0, (3)

on thesubstitution of p

for

-^,become

dx

/(l"|,P,.)=0,an equation of the firstorder in p and y. Suppose that its solu-ion

is

Then the solution of (3)is

JF(y,c)^*

Equations of the type in Art. 110 and Ex. 1, Art. Ill, are ex-amples

of the equations discussed in this article.

Ex.1. Solvea;^ +^

= 0.dx^ dx

Ex. 8. Solve ^+a"^ = 0.

dx^ dx^

Ex. 8.. Solvey^ + f^y=l.




113. Linear equations. General properties. Complementary Func-ion.

Particular Integral.

The form of the linear equation of order n is

g+i'.^î".^^-+i-^.|+i-^-1).

where Pi, Pj,---, P,^ X, are constants or functions of x. The

linear equation of the firstorder was considered in Art 102,

The completeintegral

of

is contained in the complete solution of (1). Ify=fi(x)

be an

integral of (2),hen, as may be seen on substitution, y =

Ci/i(aj)Ci being an arbitrary constant, is also an integral. Similarly,

ify=M^)y y=M^)j """" y=fn(^)"

be integrals of (2),then

y =

Cif2{^)"""" y="^n/"(")"herein c^, """, c^ are arbitrary con-stants,

are integrals of (2). Moreover, substitution will show

that

y =

"hMx) + c^/aCaj)... + cj;(aj) (3)

is an integral. If

/i("),/2("),""""/"a?),re linearly independent,

(3)is the complete integral of (2),ince it contains n arbitrary

constants.

Ify==F{x)

be a solution of (1),hen y = Y-\-F(x), (4)

in which F=c,/,(x)+ (^^{x) h c^fn(x),

is also a solution of (1). For, the substitution of Y for y in the

first member of (1)gives zero, and that of F{x) for y, by hy-othesis

gives X, Since the solution (4)contains n arbitrary con-stants,

it is the complete solution of (1). The part Y is called

the complementary function,and the part F{x) is called the par-icular

integral. Equations of theform

(2)willbe

considered in

the articles that follow.




114. The linear equation with constant coefficients and second

member zero. On the substitution of e"^ for y^ the first mem-ber

of

becomes (m*+ Pim**"^ H h P^-iWi + Pn)e^.

This expression is equal to zero if

mr + PimT"' + .." + P.îm -h ^= 0. (2)

The latter may be called the auxiliary eqaation. Therefore, if

mi be a root of (2), = e"*!* is an integral of (1); and if the n

roots of (2)be m^ m^ """, m^ the complete solution of (1)is

y = Cie-H* + Cje-v H 1-c"e"" (3)

Ex.1. Solve ^-2f?-86y 0.

The auxiliary equation is m'^ " 2 w " 36 = 0 ;

and itsroots are m = " 5, m = 7.

Hence, the complete solution is

y = cie-*" + Cjc^*.

If the auxiliary equation has a pair of imaginary roots, say

mi = a + t)3,m2 = a "

i)8 (idenoting V" 1),the corresponding

part of the integral can be put in a real form. For,

=

e*'|ci(cos)3aj1

sin fix)-f C2(cos)8a?isin^Sa?)}= e"* {A cos px-\-B sin )8a:).

Ex.8.

Solve^+8^+26y0.

The auxiliary equation is

w2 + 8 w + 26 =s 0,

and its roots are m=" 4 + 3i, m=:" 4 " 3i.

Hence, the integral is y = c-4" (cicosSx + C28in 3a;).




Ex. 8. Solve2^ 6^-12" = 0.

Ex. 4. Solve 0 + 8y = 0.

Ex.5.Solved-^-

6y = 0.

E.,.Sol.eg-3S-6|8, = 0.

Ex.7. Solve

^-a*y0.

115. Case of the auxiliary equation having equal roots. If two

roots of (2)Art. 114 are equal, say mi and mj, solution (3)becomes

y = (ci-j-cs)"""' + Cge"^ H h cê^.

Since Ci + c^ is equivalent to a single constant, this solution has

n " 1 arbitrary constants, and hence, it is not the general solu-ion.

In order to obtain the complete solution in this case, make

the substitution

wij = Wi-|-

h.

The terms of the solution that correspond to mi, m^ will then be

y = Cie"^* H- C2e^'"i+*^*,

that is,. y = "r^' (ci+ Cje**).

.

On expanding e** in the exponential series, this becomes

=

e"*i'(j-J5aj+ 1 02^W+ terms in ascending powers of h),

in which-4

=Ci H- Cg, and J5 =

c^,

Now let A approach zero, and solution (3),rt. 114, takes the

form

y =

e"'i'(-4Bx) H- cjge"-* H h c^c"^;




for the arbitrary constants c" C2 can be chosen so that Ay B will

be finite,and that c^^,etc., will approach zero. If the auxiliary

equation have three roots equal to mi, it can be shown that the

corresponding solution is

y =

e"H*(ci-fjjaj-f-Csa^,

and, if it have r equal roots, that the corresponding solution is

y = e"i* (ci-f-Cjic H h c^_iar-^).

If a

pair ofimaginary

roots,a

+ ip,at, "

i)3,occurs

twice, the

corresponding solution is

y = (ci Cja?)e"*+V""(c + C4a;)e"*-'^^",

which reduces to

y =

e^\{A + A^x) cos fix+{B + BjX)sin^Sa?}.

Ex.l. Solve ^-3^+ 3^-y = 0.

(to* dx'^ dx

The auxiliary equation is

m"-3m2H-3m-l=0,

of which the roots are + 1 repeated three times. Hence, the solution is

y = e"(ci+ Cax + CsflB^).

Ex.8. Solve ^+ 2^+*? = 0.

dofi dx^ dx

Ex.8. Solve ^~^-9^-ll^-4y = 0.dx^ dx^ dx^ dx

116. The homogeneous linear equation with the second member

zero. This equation has the form

wherein Pi, j"2,""-,/"""^re constants.

(a)First method of solution. If the substitution

z = log X, that is, a? = e*,



226 INTEGRAL CALCULUS [Ch.XIU.

be made, equation (1)will be transformed into

wherein qi, q^, ""-, g^, are constants. This can be easily verified.

Ex.1. Solve

x2^-a;^+y= 0.

dx^ dx

If " = logx, then ? =-, and

dx z

d}[_dji dz

__ldydx dz dx xdz

dx^ dz\dx)dxstî\dx dz)'

Substitution of these values m the given equation changes itinto

the solution of which is y = e*(ci+ C2"),

whence, y = x(ci+ cj logx).

(b) Second method of solution. The substitution of oT for y in

the firstmember of (1)gives

{m(m " l)-"(m " n + l)+pim(m -- !)"""(m "

"4-2) + """

This is zero, and accordingly y = af' is a solution of (1),f

m(m" l)--.(m n-^ l)-j-j"im(m T)"- (m^n'\-2)-\

+ i"n-im+i"n = 0. (3)

Hence, if m^ mj, """, m^, are the roots of (3),he complete solu-ion

of (1)is

y = CiQCT^4- c^ H h C^n,

To a solution y =

e""(ci CjS: + """ H- c,._iaf-^)f (2),here cor-responds

a solution.v

=

aj*(cih CglogajH- """ +c^-i(loga:)'"-^)f

(1),since 2 = log a?. It can be easily shown that the auxiliary



116.] ORDINARY DIFFERENTIAL EQUATIONS 227

equation of (2)is identical with (3). Hence, if m^ is repeated r

times as a root of (3),he corresponding solution of (1)is

y =

ojî^CiCj log a; -f...

+ c^_i(logj)'"^}.

Ex.2. Solvea;8^+3a;2^ +

^+ y = 0.

Substitution of x* for y gives (m^ -f l)x""0,

of which the roots are-1,

1":^ Lll2^.2

'

2

Hencethe solution

is

2/-=^+ x*{2

cosfôgx\+

C8

sin/:ôgx\|

Ex. 3.x2^+4x^

+ 2y = 0.dx^ dx

.

Ex.4.x2^-3x*^4-4y

= 0.dx^ dx

Ex.6.

x3^-3x2^+7x^-8y= 0.

dx" dx^ dx^

EXAMPLES

1. If y = Ae^ + Be-'", prove that^-m

= 0.

2. Derive the differentialequation of all circles which pass through the

origin and whose centers are on the x-axis.

3. sec2 X tan ydx-^ sec^ y tan x dy = 0.

4. xdx + ydy = a^''^y-y^.x2 + y2

5 (?x_

dy

x^-2xy~y^-2xy

6. (2ax + by-hg)dx-h(2cy-hhx-{-e)dy = 0:

7. (l-{-xy)ydx-h(l-xy)xdy

= 0,

.8.

y(2 xy + C) (?x - 6'(?y = 0.

9.

.!-",. + 1.

10. cos2 x^^ + y = tan X.

dx

INTEGRAL CALC. " 16



228 INTEGRAL CALCULUS [Ch. Xlll.

11.

X(l-x2)^+(2x2_l)y=:aX".12.

3^+ yco8x = y"8in2x.

13. ^ = x"y8-xy.

14.p8(x+ 2y)H-3i)2(x+ y)+ (y+ 2x)p = 0.

15. xp2-2yp + ax = 0.

16. p2y + 2px = y. ^4.t/-0dx8

^~

17. 6"'(p-l) + p"e"i'= 0.

(?X2

M. ^-4^1+ 8^,-8^4^=0.

dx* dx" dx* dy^

85. ^-2^ + ^ = 0.dx* dx" dx*

"" fa-")-^'S="- 29.EI^

=

-ybx,dx*



APPENDIX

NOTE A

[Thisnote is supplementary to Art. 33]

A method of decomposing a rational fraction into its partial

fractions.

Suppose that ^. (/^

"

r- is a proper rational fraction. The

substitution of a for x in this fraction, {xâf being left un-changed

and the subtraction of the fraction thus formed gives

m m_Ax)F(a)^f(a)F(x)F(x)(x-ay F(a)(x-ay'~ F(a)F(x){x-ay

'

^^

Thenumerator of

the fraction in thesecond member of (1)

vanishes for xâ, and hence it is divisible by a? " a. Let the

quotient be denoted by"l"(x).

hen

f(x)_.,

f(a) 1 i^jx)

F{x)(x-ay"

F(a)(x -

a)""^

F{d)'

F{x)(x-

a)""^'^^

Of the two fractions in the second member of (2)the firstis

one of the partial fractions required, and the second has a de-ominat

of lower degree than the original fraction possesses.

The second fraction can be similarly decomposed, and by the

repetition of the operation all the partial fractions will be

found.

When the factors of the denominator are all different and

of the first degree, the ^decomposition of a fraction can be

effected very quickly. For example, on taking the fraction

229



230 INTEGRAL CALCULUS [Note A.

7 ./

^

\.-, r^ and substituting a for x except in x " a,

(x "

a)(x" b)(x c)

^ ^

and subtracting the fraction thus formed, there is obtained

/(^) /(g)^

F(x)

(x'"a)(x^

b)(x "

c) (a? a)(a" 6)(a "

c) ("" ft){x"cf

in which F(x) is a constant or of the first degree in x.

The partial fraction whose denominator is x " a, which is

formed by thisrule,

isaccordingly ^1^21

The

partial fractions whose denominators are (x " b),(x"

c),canbe

written by symmetry. This is easily verified. For, on assuming

(x"

a){x" b){x"

c) x " d x " b x " c

and clearing offractions,

f(x)= ^ (a? 6)("-

c)-I-J5(a? a)(a? c)H-

C (a? a)(a? 6).

The substitution of a for x gives

/(a)=

-4(a" 6)(a"

c)j

whence,

A =

/(")

(a" b)(a c)

It will be found, on putting 6 for x that

B= m,

(b"a)(b-c)'

and on patting c for iK, that

c= M

(c d)(c b)

Therefore,

(a? d)(x b)(x c) (x"

a)(a b)(a c)

_

.^m+

m.+ 7^

.;""'..

"

-.+(b-a)(x-b)ib-c) (c-a)(c-b)(x-c)



Note A-B.] APPENDIX 231

Ex. 1.

2a;g-l_

2.22-1 2(-3)"--l

(a; 2)(a; 3)(x- 6) {x-

2)(2 +3)(2-

6) (-3 -2)(3 + a;)(-3 -6)

'^(6-2)(6 + 3)(x-6)

7^

17^

49

16(a; 2) 40 (aj+ 3) 24 (a; 6)

jj 23a? + 2 3-2 + 2

_"'

'

(z-2)2(x-3) (X- 2)2(2-3)

On determining F by subtraction,

3x + 2 8^

11

(X - 2)2(x - 3)"^

(X - 2)2 (X - 2)(X - 3)

11 11

(3-2)(x-3) (x-2)(2-3)

Hence,_l"+2_^

8_

_1]L-_1L_.(x-2)2(x-3) (a-2)2

X-3 x-2

NOTE B


To find reduction formula for jx*^(a + bx*^)Pdx by integra-ion

by parts. In what follows jx'^(a baf^ydxwill be denoted

by /.

(a)On

putting

dv =

Qif^\a-

boi^yx, u =

af*""+\

it follows that v = %t-^^^\ du = (m - n-i-

1)af-" dx,

nb (p-^1)

Hence, 7=

,\

^

^/". "

-Î af *(a+ 6aj")^^da?.

But aJ"-*(a 6aj")'+iar-"(a 6af")(a6af")'







234 INTEGRAL CALCULUS [Notb C-D.

From this, on solving for /,

sin^-âjcos^+ôj ,m " 1 C^-m-t,

Join

m; UUS "/,

7f* J. I

^;_m-S

^ ^^^n ^J^

=

H I Sin"*ajcos*a?aa5.

f^-1

From this result, on transposition and division by "

,

fsin"*-^cos" xdx =

^^^^^"^ ^

^^f'^'^+ ^^^^^ rsin"? cos" a: da:;J m " 1 m " 1"/

whence, on changing m into m-\-2j

*/m + 1 mH-l"/

Formula C, Art. 51, can be obtained by writing

/= Icos'*"âjsin*ajd(sinaj),

putting dv = sin"*x d (sina?), = cos"~^ a?,

and then integrating by parts and reducing.

Formula Z",Art. 51, can be derived from C by transposition and

the change of n into n-\-2.

NOTE D

[This note is supplementary to Art. 67]

It is explained in the differential calculus that if the differ-nce

between two quantities be infinitesimal compared with

either ofthem, then the limit

oftheir

ratio

is

unity, and either

of them can be replaced by the other in any expression involving

the quantities. A deduction that can be made by means of this

principle is of great importance in the practical applications of

the integral calculus.

Ifai-f-a2H ha"

represent the sum of a number of infinitesimal quantities which

approaches a finite limit as n is increased indefinitely.



notbD-b.] appendix 236

and if jSi,p^....,

^"

be another system of infinitesimal quantities, such that

where ei, eg, """, e"

are infinitesimal quantities, then the limit of the sum of Pu p^t

""', Pn is equal to the limit of the sum of "!, Oj, """, a".

It follows from equations (1)that

A+i82+ - +Pn=("î+^+ - +a")+ (aiei+a2e2+ +anen).(2)

Letrj

be one of the infinitesimal quantities Ci, 62, """, e,, which

is not less than any one of the others. Then

Wl+

P2+- +

Pn) ("!+ 02 + - +

"")"("!+ "2 + - +

"n)1/-

But by hypothesis "! + 02 H- """ + "n 1^3,8 finitelimit, and hence

the second member of this inequality is infinitesimal. There-ore

the limit of jSi+ ^82+ """iî.is the same as the limit of

NOTE E

[Thisnote is supplementary to Arts. 84-87]

Further rules for the approximate determination of areas. A

few more rules for approximately determining the area of

LAKT (Fig.63) may be stated. As before, h denotes the

interval between successive equidistant ordinates, and merely

the coefficients of the successive ordinates are given in the

formulae. In the trapezoidal rule, strips were taken in which

two ordinates were drawn; in other words, the ordinates were

taken by twos. In the parabolic rule, strips were taken in

which three ordinates were drawn; that is, the ordinates were

taken by threes.

* See B. Williamson, Treatise on the DifferentialCalculus^ Arts. 38-40.





Note E-F.] APPENhlX 237

found by integration. For example, in the case of each succes-sive

pair of ordinates in Art. 85, the given arc was replaced by a

straight line, and in the case of each successive group of three

ordinates in Art. 86, the given arc was replaced by the arc of a

parabola. On assuming that the equation of the second curve is

y = A +-4iaj

+ ^2aj'+-+A"", (1)

the coefficientsA^ A^, A^ """, A^, can be determined. For, the

substitutionin

(1)of the coordinates of then

+1

given points,

namely the extremities of the given equidistant ordinates, will

give w + 1 equations, by means of which the values of the n + 1

coefficients A^ A^, """,A^, can be found.* If n is sufficiently

great, the difference between the area of the second curve and

that of the original curve will generally be very small. The

general

formula for the case

ofn + 1

equidistant ordinatescan

also be deduced by the method of finite differences.!For a dis-ussion

on various methods of finding an approximate value of a

definite integral by numerical calculation, reference may be made

to J. Bertrand, Calcvl IntSgraly Chapter XII., pp. 331-352.

NOTE F


The Fundamental Theory of the PlanimeterJ

In Fig. 58, ALBO is a plane figure whose area is required, and

QX is a given straight line taken as the axis of X Let MK rep-esent

a plate of which two given points always move, Q along

QXy and P on the contour of the given area. Then QP is a

straight line fixed with reference to the instrument. Let b be

the length of QP. Let W be the recording wheel with axis

parallel to QP. Its actual location is arbitrary.

* Also see Lamb, Infinitesimalalculus^ Art. 112.

t See Boole,Calculus

ofFinite

Differences hapter III,Arts.

10-14.

X This note is by Professor W. F. Durand, who has kindly permitted its

insertion here.



238 INTEGRAL CALCULUS [Note F.

The movement of P from P to G, a point very near, may be

decomposed into: (1)A movement dx parallel to QX, (2)a

movement dy at right angles to QX. It will first be shown

that the record of the wheel W due to the dy component will, for

the closed area, be zero.

Fio. 68.

It may be noted that the amount of the dy record depends on

the dy and on the configuration of the instrument under which it

is traversed. Now it is evident that for every dy traversed in

the up direction, there will be an equal dy traversed in the down

direction, and under the same configuration. In the diagram the

pair thus traversed is dy and dyi. The net record for such a pair

is zero, and for every other pair, zero, and therefore, for the

entire contour, zero.

It follows that the entire record will be merely that due to the

dx components. This is found as follows. The component of

dx in the direction of the plane of the wheel is dx " sin 0, But

sin^ =^. Denote that part of the record diie to dx by dB.b

Then,

dB=dx'sm6

=_ydx



Note F.] APPENDIX 239

Hence, E =

-(ydx=z",

and therefore A = bB.

It only remains therefore to graduate W conformably to the

length of 6, or, vice versa, to graduate W and give to b an appro-riate

length. The latter is the usual method. By giving to b

various lengths, the area may be read off in corresponding units.

Thus far it has been assumed that Q follows the straight line

QX, It will next be shown that the record is independent of the

path of Q so long as it is back and forth on the same line.

Fig.

To this end let FM (Fig.59) be any area, and ABODE a

broken line. Iêt A and E be the points from which arcs with

a radius b will be tangent to the contour at F and M, With the

same radius and B, C, D as centers, draw arcs as shown in the

figur.e. Suppose now that P is carried around these partial areas

successively, and always in the same cyclical direction. For

(1)the point Q (Fig.58)will traverse AB, for (2),C, etc. In

each single case the record will represent the corresponding area.

Therefore the total area will be represented by the total record.

And it is readily seen that GH, IJ, KL, are each traversed twice

in opposite directions. Hence the record due to them is zero,

and the actual record is due only to the external contour. Hence,

if P were carried directly around the external contour, it should



240 INTEGRAL CALCULUS [Note V-G.

have the same record, and hence the area. This is true for any

broken line,and hence for a curve. In the common polar planim-

eter the curve is the arc of a circle. Thus the point O in Fig.

55, Art. 88, which corresponds to the point Q moving along QX

in Fig. 58, or along ABODE in Fig. 59, moves along the circum-erence

of the circle of center T and radius a.

NOTE G

On Integral Curves

1. Applications to mechanics. " (This article is supplementary

to Art. 93.)

(a)The statical moment about OT=M^ = (area)H= ayiOH.

Hence,

M^ = ayi (aJiHX) = ay^x^ - ahy^ = a {y^x^ hyi), (1)

Also

Igh = Ipx - (area)HXf = 2 obey,- ?^^ =

ah(2y,-^\ (2)

(h)The value of HX in Art. 93 (6),ay be found by a simple

construction, though from its nature the accura"5y may not be

all that is desirable.

Let BH be drawn tangent to OB at B. Then

tanB^X = ^=-^dx HX

But 2/2-

Gidx, and hence ^=

^.hJ dx 0

Hence,

-^=^,and HX =

^^HX 0 yi

Hence, from Art. 93 (6),he point H thus determined will be

the abscissa of the center of gravity as desired.

(c)The moment of any area ORFH about any vertical XA is

proportional to the corresponding ordinate XB of the tangent



Note G.] APPENDIX 241

to the second integral curve at the point E on the limiting ordi-ate

HF,

It has been shown in Art. 91 that-^

=

-^pFrom this,

dx b

tanZ".0: =^=^^

=2^ârea,

HK dx b- ab

Hence the equation to the tangent KD is

at)

whence, aby = ab " HE-{-A(x-

OH). (3)

But from Art. 93, ab . HE is the moment of the area about

HFy and A{x " OH) is the correction necessary to transfer this

moment to an axis distant {x " OH) from HF, and therefore

distant x from the origin. The second member of (3)is thus

seen to be equal to the moment of the area about a vertical line

at any distance x from the origin. Hence, such moment is meas-ured

by aby, or ab times that ordinate of the tangent line which

is determined by the abscissa x. Hence such ordinate at any

point bears the same relation to the moment of ORFH about

the vertical line containing the ordinate, that HE does to the

moment about HF.

(d) It follows that where KD crosses OX, the moment about

the corresponding ordinate will be zero, and hence an ordinate

through K will contain the center of gravity of the area ORFH

Hence the construction given in (b)above is a special case of (c).

(e)If we apply the same proposition to the moments of the

two areas ORFH and ORPX about an ordinate through L, the

point of intersection of the two tangents at E and B, we shall

have for each moment the expressions abNL, and the moment of

the difference of the two areas or of HFPX about NS will be

zero, and therefore NS will contain the center of gravity of such

area.



242 INTEGRAL CALCULUS [Noxa G.

Hence the tangents to the second integral curve at any two

ordinates intersect on the ordinate which contains the center of

gravity of the area of the fundamental curve lying between the

two ordinates chosen.

2. Applications in engineering and in electricity. The limits of

the present article will not allow detailed reference to the various

ways in which these curves may be made of use in studying

engineering problems. A few brief references may, however, be

made to some of the more common applications.

It is readily seen by comparison with text-books on mechanics

that if for the fundamental we take the curve of net external

force on a beam or girder, then the first integral of such funda-ental

will give the entire history of the shear from one end to

the other. Also that the second integral will give similarly the

entire history of the bending moment from one end to the other.

This serves to illustrate one important advantage of representar

tion by means of these curves, and that is, that they serve to

give not only the value at some one or more desired points, but

at all points as well.

In this way they furnish a continuous history of the variation

of the function in question, and thus give a far more vivid

picture of its characteristics than can be obtained in any other

way. In the case of beams or girde^s it may be well to note

that external forces should not be assumed as concentrated at a

point, but should rather be considered as distributed over a length

equal to that occupied by the objectto the existence of which

they are due. Thus the supporting forces at the ends of a bridge

span must not be considered as located at a point, as is common

in the analytical treatment, but rather as distributed over a length

equal to that occupied by the supporting pier. Their graphical

representation will therefore be a rectangle, or rather it may be

so taken for all practical purposes.

As another application, consider the action of a varying effort

or force acting through moving parts having inertia, and upon




a dissimilarly varying resistance, their mean values being of

course the same. This is the case with the ordinary steam

engine or other prime mover operating against a variable resist-nce.

Suppose that we have plotted on a distance abscissa, the curves

of effort and of resistance. The integral of the first will give

the history of the work as done by the agent or effort,while that

of the second will give the history of the work as done upon the

resistance. Steady conditions being assumed, their mean values

will be the same. Their history, however, will be quite different.

The difference between the ordinates at any point will give the

work stored as energy in the moving parts during their accelera-ion

when the effort is greater than the resistance, restored

during their retardation when the effort is less than the resist-nce.

We might reach the same results by taking as our funda-ental

the difference between the curves of effort and resistance.

The integral of this will give the history of the ebb and flow

of energy from and into the moving parts of the mechanism.

Again, by replotting this latter curve on a time abscissa it becomes

representative of the time history of the acceleration of the

moving parts.If then the

reducedinertia

ofthese

partsis

known, the acceleration at any instant is known, and the curve

may be considered as one of acceleration. Its integral will,

therefore, give velocity, such velocity being the increase or

decrease above the mean value. Such a curve would, therefore,

show the continuous history of variation in the velocity due to

the causes mentioned.

In electrical science there are many interesting applications of

these methods. Of these only one or two of the simpler will be

here given.

Suppose that we have on a time abscissa a curve showing the

history of the electromotive force in any circuit. Then since

this is the time rate of variation of the total magnetism in the

circuit,it is evident that, reciprocally, the latter must be the

INTEGBAL CALC. " 17



244 INTEGRAL CALCULUS [Note G.

integral of the former; Hence the first integral curve will give

the history of the total magnetic flux in the circuit.

Again^ if we have on a time abscissa a curve showing the

history of a current, then the history of the growth of the quan-ity

of electricity will be given by the first integral of such

curve.

Instances might be widely multiplied, but enough has been

given to show that where desired results may be found by one

or more integrations effected on a function whose history is

known, the complete representation of the problem naturally

leads to the production of these curves ; and for their practical

determination and for their application to many special problems,

the fundamental relations and properties as developed above and

in Chapter XII., will be found of considerable value.

3. The theory of the integraph. We will next show briefly the

fundamental theory of the integraph, an instrument for practi-ally

drawing the first integral from its fundamental. Various

forms of instrument have been devised, but in nearly all, the

kinematic conditions to be fulfilled are the same. These are as




follows : Let PC (Kg. 60) be the fundamental relative to axes

OX, OF; and QD the integral relative to axes QXi, QY. For

convenience the two Faxes are taken in the same line, though

this is not necessary. P and Q are therefore corresponding

points. At Q draw a line QA tangent to QD, From P draw

PE parallel to QA. Then ;

OA dx OE a

Hence, J^ ôr y, = pcto.

dx a^

J a

If now a is constant, we shall have

1 /*" Jarea

yi = - \ydx=z or area = ayi.aJ a

These conditions are seen to correspond to (2),Art. 91. The

instrument must therefore include three points, E, P, and Q,

related as above specified. While the instrument travels along

th(B direction of X, P is made to trace the given curve, and E

remains at a constant distance a from the foot of the ordinate

through P, This determines a direction EP, and Q constrained

by the structure of the instrument to move always parallel to

EPy will trace the integral curve QD.

It is not necessary that the points E and A should lie to the

left of 0 as in Fig. 60. They may be taken as at E^, A\ and in

such case if the fundamental lies above X, the integral wUl lie

below its X as shown by QZ"',and vice versa. The actual values,

however, will remain unchanged, and the inversion is readily

allowed for in the interpretation of the results.



Following are the figures of some of the curves referred to in

the preceding pages :

The hypocycloid, ac^+ y^ = a^. The cissoid, y^ =

2a-x

o O* X

The cycloid, x = a(^" sin "?);

y = a{\"

cos^).

246

Folium of Descartes,

aj8 + y" = 3aa;y.



X X

The catenary, y = ^(c*+ e ").

O a

The parabola, x^ + y*= a*.

The semicubical parabola, ay^ = ic8. The cubical parabola, a!^= "*.

The parabola, r = a sec^ -.The logarithmic spiral, r = e"*.

2

247



The curve, r = a sin' Spiral of Archimedes, r = a0.

The curve^ r = o sin 2 "?..

The cardioid, r =

a(l- cos 0).

The lemniscate, j-^ = a^ cos 2 0.

248

The witch, " = "

i^f!".



A SHORT TABLE OP INTEGRALS

Following is a listof integrals for reference in the solution of

practical problems. The deduction of these integrals will be a

useful exercise in the review of the earlier part of the book.

GENERAL FORMULAE OF INTEGRATION

1. j(u"v"w "'")dx

= iudx " ivdx" iwdx" """.

2. Imu dx^miudx.

3 (a), iudvûv" ivdu.

Z{b). Cri^dxuv-Cv^dx.dx J dx

ALGEBRAIC FORMS-

4. r^^^logo..J x

a;" dx =

-, whenn is different from " 1.

Expressions containing Integral Powers of a+bx

7. C(a+ bxy dx =

(^+

hxy-^"^^^^^^ ^ .^ îigej.gjj^ ^^ _ ^J o(n-\-l) .

249



260 INTEGRAL CALCULUS

8. IF(Xj a + bx)dx. Try one of the substitutions, z = a-{-bXy

10.

" +

a^dx 1'" J^^ = J[K""")*-2a(a + 6a,)+a'log(a 6a;)].

dx 1,â-|-6a?1 r dx

^Îq

Jx{a-{-hx) a x

J x^(a'\'hx) ax or X

16. r - =^ iiog5^"^.

Jx(a+ bxy a{a+ bx) a* a

J {a+ bxf b\ a-hbx 2(a-{-bxy]

Expressions containing. a^ + a^, a^ " a^y a-i-baf, a + b7?

18. r_^^=J_iog?L""; r_^_=iiogî^.J a^

^QcF2a a " x J a^â^ 2a

,r-!-a

19. r ^^

=-l_tan-â^v/-,hen a"0 and 6"0.

Ja + ftaJ* Va5^"

20. C ^^

=J-loggHLJg,



A SHORT TABLE OF INTEGRALS 251

21. Cixfîa-^-hotfydx

"(ripH-?n l) 6(Mi" m-f-iy

^

22. CQir{a'\-lxxfydx

np H- m-f

1 np -I-m H- !"/

23. f- *"

(m" l)aaf"-\a4-"a^/~^ (m-l)o J"""(aH-"a;")''1

(m" n+np" 1)" /* cga;

24. f-"^

1,

m " n + np" 1 /^ da;

""

an (p - l)af-^ (a+ 6af)^^ a7i (i" 1) J aj""(a+ hx^y-^'

J af

a(m" l)af-* a(m

"

1) J af"*

2g f(a+6af/(^J af

_

(a^hafy anp r(a-\-bafy-^dx

(np" m 4-l)af~^ np

"

m+l*âf

21^ r ic^diB

'"/jr6ic"/af*-"+^ a(m

" n-\- 1) T a?"*-"da;

6(m - np -f-1)(a+ bx^y~^ b(m - wp + 1)J(a+ """/

af*da;

(a4- ba^y

af-^^m + n " np + 1 (* a;*"dxIf

an(p- 1)(a+ 6a;"y-^ an (p-1) "/ (a+ 6aJ")'




29. f" x

as*)"

2 (n - 1)a' L(a" a^-'"*" ^

" ^V

(o"+a^" J

30; f" ^J (a+ 6aO"

"^

2(n-l)aL(a6a!")"+ (^" ~ ^)J(o+

fta^"31. r_^^ = ir_^,wherez=:"'.

J(o + 6a^" 2*/ (a+ 6z)-'

32 r_^^_J (a+ bot^'

"

" "

1 C dx

2 b(n - 1)(a+ 6a!")"-"2 6(n - 1)J (a+ 6a!")"*

33. f ^=J-log

"^.

Jx(a-\-baf) an a-{-baf

Ja-^-bx' 2b^\ ^b)

3fir ^dx

_x

a r dx

37. f^^

=^log

^.

38. r ^=,i"

^r ^.

J7?{a-{-by^

ax aJ a-\-ba^

dx39

(a + "a?2)2 2a (a 4- ""a?^ 2aJ

a(a+ "a?2)22a (a4- ba^ 2aJ a + fta?




Expressions containing -Va + bx

[See FormulaB 21-28]

40. rxV^+todx=

-2^"ri3MVi^"M..

' loir

J^

106 6*

44. f '^'^

=JLlogV^+^-V^.fora"0.

45. r_^=, =

_2_tan-'J^"^,fora"0.

*^ arVa H- fta? V" a^

" a

46r c?a;

_ -~Va-f-6a;"_ T d

47. rV^^T^^^2V^T6^+ ar"

jJ X J

xVa

dx

4- 6a?

Expressions containing â? + a*

[SeeFormulae 21-28J

48. r(aj"a")ida?|V?T^" + ^loga?4-^s^T^.

49. r(a!"+o")*cte= - (2a?+ 5a")

VF+a* + ^log("+ a/^+o").

n

J w + 1 w + U



H+2

2


H

52. ra!"(!r"a')ida!==|(2a!"a")V?Tc?-^*log(a!+V?+

"/o o

'

(to53. f '^

=log(ic+V^+T").

54. f_-^_ = /" "

-'(sr^ a*)*2 2

r_^ = llog ^=.

66. r-^^=v^+^.

56

57

5S.

69.r tto

^

Vig'+ o'

60 /dx"_

Vaj'+ g", 1

io"a+Va!'+'0'

a!"(a!"a")i2aV 2a" a;

"/Of

a;

^"a!"

62.

ri^"#^- ^^^"^+ log (a, V^+Ô.




Expressions containing v^--^

[See Formulae 21-28]

63. f(a^-aîda;?V?^^-f log(a; V^^^.

64. r(a!'-a")*(to=|(2!e"-6a")V5^^=^?|^log(a;+V?^."/ 8 o

n

n

66. ^x(x" a')*iB= MufQ.

67. fa!*a!"o")*to= |(2" -

a")V^^=^- 1'og (x+ V?^:^).

68. f ^". = log {x + V5^"=^").

'^ (!r"-a=)*

2

/7

da;

70. r^^=v?:r^"-

71. r-^^ = |^?3^"+ |'log(a"+V^^rp)."^ (""-ay

2 2

72. r-^^ =

-^" +log(a;+Vx^^r^')."^ (a'-a')* Var'-a''

73. f ^5_=

lsec-'*;f '^'^

=sec-"a;.

*'a!(a^-a")5" "

-^ asVa^-l

74 /^ da;_ -y/g' a'




mmP dX

-v/aj* (3^2 \ Qg75. I irr

= jLJi"

^ 4-_:!^gec-*-.

76.J X X

"^ J(^-J-)^^^,V^+log(,V^3^')"

Expressions containing -s/a^-ôi?

[See Formulae 21-28]

78. r(a^-"^*(foj^VS^:r^+f8in-i?.

"/o 8 a

80. r(a'-aîdx ^("'-f)'-^

fCa'-aO^'dxJ^ '^

n + 1 w + lJ^ -^

11+8

81. fgCa"x^*dx

=

-("'-'^j .

J^ '

n+2

82. C"^{d'-aîdx=^(2!i^-a^-y/a^-x'J o 8 a

83. r_^_=8m-"*; f" ^"=sm-'".

rtto_

g

- (a*-a!")*'~aVa*-a^

84

85

86

-'(a'-af)*

87. r-^_=^=^_sin-




88.r a^-c?a. g!:^^^,3^^(m^l)aY-'^^

89. f^^

=llog ^=.

f"" ^?_=.A I t"t"_

Va^ " a*

a*a?

91. r_^^ =

_^g^+^,log

J X"

X

'-a?

93.

J(5^z;A*da,^-^5^^--'*

Qv*^ "

" ^" ^^^"

oxu." "

or X a

EXPRESSIOKS CONTAINING V2 aX " Q^, ^2 OX-\-0^

[SeeFormulae 21-28]

94, C-y/2ax-a^clx5-=it V2 ooj - ar^ + ~

vers-^^

"7.22 a

96. f^^

=yers-^g: f_^=, = vers-^ a?.

96. I af'v2aa? " a:*cte=^^

"

^"

^

m-\-2 J

97r t?a;

_

V2 ax " a^

"â^V2aa?-aj2""

+ -

(2m - 1)oof"

i " 1 /" c?a;

i " 1)a J a?*-*2m - 1)a Jaj"-* V2aa?-aj2




Qfi C ^^=

arW2ax"

j^(2m" l)ar oT-^dx

J"- (2m-3)aaf (2m-3)aJ a^-^

*

00. I a?v2aaj" ar^cfa;= -^---v 2 oa;" aj*-f-"vers"^-.

J 6 2 a

01 r cga; ___ V2 ax

"

g*'

"/ "V2aa;-a2oa?

02. f ^^==-V2aa;-a?^

4- gvers-^g.J ^2ax-d' a

03. r-^^^=, =

-5Jl5"V2^^Z^+

3^2êrs-^f.

*^.V2aa;-a^2 2 a

04. r^^^~^da?=V2aa;-a?" + a vers-^g"

J X a

J ar^ a; a

jjg/*V2 oa; " ar

^^ ^ _

(2aa; a^^f

07 C- ^^"

a? " a

(2aaj-af)f "V2aaj-a^

08(2aaj-ar^f "V2aa;-aj*

09. iF(x, "yj2ax"x^dx= jF(2;+a,Va^" 2*)a;,where 2=a?" a.

10. f^^

=log(a?4-a4-V2aa?4-a^.*^ V2 aa; H- ar'

11. ji^(aJ,2aa;4-ar^)^x=|F(z"a, -y/z^"a^dZyhere z=x-\-a.




Expressions containing a-\-hx "cai?

112.

f ^L-- ^^tan-"-|^"L,wheii6""4J a + bx + ca? V4ac-6* vToc^^

113. =

1log

2ca" + 6-Vy34^^^^^^

V6' - 4 oc 2ci"+6+V6'-4ac

6""4ac.

114. r__^_=

^log Vy + 4oc + 2ca;-"

Ja-\-bx-csi?y/h^jf.4:w: V6' + 4ac-2ca! + 6*

115. f,

^^- = ^log(2cxb + 2VcVa+W+a?).

"

"Va + bx + a? Vc

116. fVa+to + cx'da!

=^^"JVa+6a;+ca!'-^'~'*"^log(2a!+6+2 V3Vo + bx + ""*).

*"' 8 c*

117. r ^

=J-8in-".^"'-ft-."^ Va + 6aj" ex* Vc V6' + 4ac

118. CVa + bx " ca^dx

4c 8cf V6" + 4ac

119. r ^'^^

^Va

+ 6^+^_6l^g(2cirft+ 2V^Va + to + caO.

"=2c'

120.r iPtfag

_

^/a+ bx-ca^

^

b

^^^_i

2cx-b

^ Va + fix" ex*c

2 c*V6'+4ac

Otheb Alqebbaic Expressions

121- /VSJ*" V(a+x)(6+x)+(a-6)log(Va+"+ V6+5).

122. rJ|^cix = V(a -

x)(6+ x)+(o +

6)sm-"A^"|INTEGRAL CALC. " 18



28i) VfTi"SLlL CALCULUS

"XFr"y":^TlAL AXD TRIGOXOMETRIC EXFKESSIOXS

"/loga

"/a

137. iffflx^ff, 1S9. |sinxdx ~eo6"

130. I"'/wixdx s sin;?.

131. CiAnxdx= logBecx^ " logeoBx.

133.Tcot

?/la?log

"in".

133. Cm'.x(Jxf ^^^-= log (sec? + tana?) logtan /"'+ ^\

"/J co" a?

\42/

134. Inoftpc? r/aj T-." = log (cosec " cot a?)

log tan ^"J J sin a?

^2

135. jHm!"ajrfaj"itan". 136. jcosec* a:da:= " cot a?.

137. jseotan xdx^ sec 0.

138. ronnnn a; cotirdx rs ^ 0086O 0.

130. rmu"("(fuj!!--!8in2a.J 1

2""4



A 8H0BT TABLE OF INTEGRALS 261

140. rcos*a? cte = ^+ -. sin 2 a?.

J 2 4

141. r8in"xda.-?i5!::^^^2s^

!^^^ fsin^-^xda,J n n J

142. rco8-"(te=522r^^siM+ !L=J:fcos-'arcto.

J'

n n J

143r ^^

__

1 coso;,

n " 2 r dx

Jsin"

a? w " 1

sin*""*; n " 1 J

sin"~*a;

144 C ^^"

1 sin a; n " 2 /* da?

"/cos* a? n " 1 cos*"* a; n " 1 J cos"~*aj

145. rcos'*a?8in"a?cfa;^^^'" *a?sin*+*a?_^m---l Tcos-^âsin-acfa-.

146. I cos* a?

sin*a? da?

sin*"*in*-* a; cos^+â;,

n " 1 /\^""^

"^,."-"^^^

1 I cos" X sm* 'X ax.

m-\-7i m + nJ

147.' ^

'"/;m"'a;cos*aj1 1

,m4-n

" 2/* dx

n " 1 sin"** x cos*"^ a?

4-n-2r

n " 1 J sii

148. f^_^Sin*" X cos* a;

___^

1.m

+ n " 2r da;

m " 1 sin**~*a? cos*"* a? m " 1 Jsin"*"^ cos" x

"

^grcos*"

xdx__

cos'"'^* a;

m^n-^2rcos"*a?da?

"/ sin" a? (n " 1)sin"~^ a? w " 1 Jsin""*"

j^gQrco8'*a;da;__ cos**-â?

. m~l /^cos*"~'a;da?

"/ sin"a? (m "

n)sin"*a? m "

nJ sin" a?

151. fsincos" a; da? = - 525!!L".

152. I sin*a? cos xdx =sin"+â;

n + 1




153. rtaii-xd!r^^^^-J'tan-*"(to.

164. CeoV'xdx=:-^^^-fcoV-*xdx.... r- "

J 8in(TO+ n)a!, 8in(TO" n)ir

^^^r -, sin (m + n)a? . 8in(m

"

n)ag156. I cos mx cos nxdx==

-" ) ;" ^ +

-o^-"

-f"J 2(m + n) 2(m "

n)

^.^

/*,, cos(m4-w)ar

cos (m "

n)"157. j8mmxco8nx"to-

^^^^^^^-

^^^_^^^"

158 r ^=

,^tan-*/" tan 5\ when a " 6.

Vft"

atan|

V6+a

159. a ^ log; , when a"b

Vft*-o' VftHatanf-Vft+a

atan|6

160. f ^= tan-t "

-iwhen a"b.

J a + bsinx Vo"-6* Va*-?**

^log

, when a " 6.161.

y/b'-a"atan|+"+V6*-o"

162 r ^=JLtaii-^/^^*?5L^Y

J a*cos*aJ + fe^sin*a? aft \ " /

" A"" r -,"

J ^(a sin nx " n cos wa?)163. I ^ sm waj da? =

"A" "

^;

/'. ,

e* (sina; " cos a?)e* sm X daj= "

^ ^"

"îi/*--" J e*^(wsinwajacosna?).164. I e^'cos nx dx = "

^^

rr^'5

/", e* (sina? + cos a?)

"* cos xdx ="^ ^

^-



ANSWERS TO THE EXAMPLES

CHAPTER n

Art. 12

8. y" = c6"; y" = "^. 4. y" = *(a^+ "J").

CHAPTER ni

Art. 18

6* 11'

TO + n + l* 100* 22' 38*

-8 3 1 1 1 1 1

40 69fi 93fi TOX* 99x" 20a;"

6. i"t. i,", !"*,

^"^', {xî"*

-^"-^-*\7. }x*, 2a;*, -

-?='***' **^' ***" ^**

vx

8. log^ log(" 1), I log(x"4- 3), \og(uvw+ 1),logCx"+ 2 x" - 2 x + 4),

log tan X, log sin X.

g 8"15 (m + n)"

10. " co8 2x, Bin3x, tan4x.'

log 2' log(w"+n)* 11. secix, 8in-i2x, Bin-i3x, sin-ûv.

12. tan-i2x, tan-i3x, Bec-i2x, sec-i4x, eecâ

Art. 19

11 d V2 + I

8. Jx8~x2 + 6x, Jx*-2x* + 2xi263



264 ANSWERS

a 2 p..

- sinmx_

coaSg tan2x_cot(t"+ n)x

m 3'

2 i" + n

7.-llog("

+ â;"),log(4 + 3ii"),-ilog{6-23B*).o

""

J-^-'.i*--^"'-''-tArt. 20

8.K^J+V,l("+

o)^log(x+ a), 2V2ri^,

1-,A{2 + 3a;)*

5x + a

-A(3-7x)^.9. sin (x + a),

tan (x + a)," Jtan(4 " 3x), "

-cos (a + to).0

10. 2Bin|,""+*",-3e"l,-

^

2sin"x

11. A(4x-3a)(a + x)*,J_(a

+ 6x)'*(26x-3a).

a

12. log tan-i X, sin log x, " n cot- .

n

18.-L("

+ 6")",A("

+ to)*,X(a

+ 6y)".

14. i8in-i2^.15.

-Jcosec^x,isin6tf-isin*tf |Bln"^+ V"n*^+2sin^, Jtan*0

- }tan80 + tan^0 + 9tan0.

Art. 21

6.xsin-ix

+

vT319. 12. a^jLltan-i x-^

6. X COt-l X + i log (1 + "'). ," r/i NO "i . o-i"^ ^18. x[(logx)2

" 21ogx + 2].

7.a'i"

^] .

14.8intf(logsintf 1).

I log a (loga)2i 15. tanx(logtanx - 1).

'22x

.

.21 iA

"*i

8.

a"{-^--2-^^+--^}.W-

j[aoga:)"-ilogx+ H.

i log a (loga)2 (loga)8i 4

9. X tan-i X - i log (1 + x^)17. J-

c"" ("ix- 1).

10. 2C08X4- 2xsinx " x^cosx.^.w+i

/ j \

11. x^sinx + 2XC0SX - 2sinx.*

w + l\ m + 1/



ANSWERS 265

Art. 23

6. log(2x-6 + 2Va;2-6a;). 7. log (7a; + "/7V7 x^ + 19).

8.

-^log (3a;2 + 1+ V3 VS x* + 2 x-i - 1)

2\/3

e.

ilogf^- 21.

-i-log^^^-Â

10. 58ln-if^^:^Vr 5vers-i^.^ , " ,/ ir\

V 2 y 2 22. |tan2x + logtanfx + M + a;.

"-'-"^23.

Isec-x--^-.

18. 2 log sec 8 X.

. /^^26. ilog(x2+v^^5nr^).

V6 V326. |(x2-y*)l

15. J-tan-i-^. / P

VS V5. 27. log(/3+V^ + 2V3).

16.

_l-logl^.28. J-vers-i^

4V2 y+Vs Ve ^

17.-log sin ox. 29. 4tan"i^-^"

18.-logsin((rx

+ 6). 3^ log tan ? + log sin ^.

19. JLtan-i^.1

,

VS VS 81. - log (a2x+6+o"/o'^"H2 6x+c).a

a* " 220. itan-i-^.

^^ ^^^^__^^^^^^^^^__^

Art. 24

8. ^-2x2i/~2xy2^.^4.c..a2x-^-a;j^-a%

c.

3 o "

6. ax2 + 6xy + ay2 + ^x + ey + *.

Page 55

1. "exs x-^", C^ + ^^-^-n (lyA bh^m + n + 2 \3y

2. Jx8 + 4x'--"x*

+ 16x*, iJ-H^J + î^.



266 ANSWERS

8."^-2^+2a:2-8x261og(x + 2), ^+ ^r^ + 6" + 51og(" - 2),4 o 6

24 + 6 log 3.

-?^x"^, _l_a;l-^h^-âv^ 6avi.

l + " n " I 1 " " 5 4

5. " 2 cot 2 tf, f sin 2 0 + ^ cos 30, _

1 log (a + 6 cos\^).

6

6. log(y2-a2), sin a; + tan a:, iClogx)2,Pog (""+ ")?.

2a

7. 106, i, 1, 4, :, log2. 8. 2, 1, i-a,"i-"-i-

4 4 2 2 a

". i^. iog2.

-0o|2)!,?_:,l(eT_,?).

o a

10. 4\/S, ^(-l + 3v'3-2V^), 3+VI6.

12.siQ-ihtl^^ 2tan-iVTTi. 18. fx*-J"*

5?!-zi"v^f+li^.

\/l3

3 62

14. X sec~i a; " log(a; Vx^ " 1), x cosec"^ " + log(x+ Vx-^ " 1),

iccos-ia;~Vn=^, ^sin-ijck*^+ 2)Vr=^.

16. as sin " + cos X, "

.'B^cosa+ Sx^sina; + 6"cosa; " 6 8inx,

x^

X tan " + log cos x "

16. a'[^ 3^_ + _?^ ?_1 cosa;(l-logcosaj),Lloga (loga)2^(loga)8 (loga)*J^ ^ ^'

cota;(l logcotx).

17. (a;2 2 a; + 5)sin (x2 2 " + 6)+ cos (x^- 2 a; + 6).

18.(^ + "^')[2log (x8+ a8) 1]_

a8(aj5 a^)[log(x^+ a^) 1].6

19. " ^|(logx)2llogx + ?}. 22. tanx-cotx + 21ogtanx.

33.*t 3 9)

23. ^ 1 log (1 + cot 0)+ ]log (1 + cot2 6).

24. sin-^^

:"26. 0. 28. 1

sec-i^

-

1 log (x + Vx^ _

a^).

26.

rtlogtan^l

^Wftlogsintf.9. i{log(x+ VxM^)F.

27. 8in-i^^=i, sin-i^+-^.80. log sec f 4-V

V3 3 V2 4;



ANSWEB8 267

81. log ?^2ii. 87. 14 sin-i5 - (f" + 10)VJ^^.C08" ^ 2

82. }"*vers-i-+f("+4a)V2a-". jg^ 1 logS^c^-^*

2V3 sec" + V3

83. ^V3fiI""^\og(x+y/s^"c?).'

r r"".

-^log^-4.1

log^-^ 2V5 e" + V5

1 //". "

-irx40. sec-^-^*

85. " logtan(f|V Vftloga V6

\/2 \2 o/

86.

ilogtaii(0+^y41.

2V28in-i^V2sm|42.

^tan-i

2"^+^. when 62" 4 ac,

v'4ac-6* V4 ac - 62

:log

2ax + 6-V6^^[I^^wheii6a"4ac

V"2-4ac2ax+"+V62-4ac

48.-Llog

(2oic + ft+ 2 Vo Vox^ + 6x + c).Va

44.-Lsin-^

2qg;~"^ ^ sinajcosy + c.

Va VdM^Toc

47. 3c8 + 3a;2y + 4a^2 + 2y" + c

46.-cosxcosy + c.

CHAPTER rV

Art. 29

2. (a) 74|; (6) f. 7. 251.l^- i*^'

8. (a)A;W4. ". f"/I^. 18. |.

6. 28H. 10. 24.^*- *^^â

Art. 30

5. 81^ ir.

6. (a) ^îr; (6) îr; (c)t^tjt;

iH"-

7. (a) fir;(6) 2ir; (c) îr; W îr.

8. (o) 47r; (6) iro21og(l+V2).

9. firc^xii 10. ^c*xA 11. ^- 18. t'Oft^-



00


Oft C ^^"

^ ^V2aa? " g* (2m " l)a/* oT-^dx

J a;" (2m-3)aa;-^(2m-3)aJ x^'

a;v2aa:" ar*da;= -^-r v2aa;" 0^*4-7:vers"^ "

6 2 a

01 r__Ê__ = " V2 ax"

g*

.

r g-^ "

==-V2aa?~ar'+ aver8-^g"

J"\J2(XX'-Q^

"

03. f" ^^= =

-î^v'2^^=^

+ |a^vers-^^-'V2aa;-a^

2 2 a

04.rV2aa?-a^^^^V2aa?-ar^

+ avers-^g"Jo: a

05. rV2""z:i^,to-2^^^Ê?-ver8-'2.a? X a

3aa^

Q.7r doj

_

a; -a

(2aaj-af)f "V2aic-a^

08.

r_^^_="

-^"

"^(2aa;-ar^taV2aa;-iB2

09. jV(",2ax"ix^dx= CF(z+a,Va*" 2*)2, where 2=0:- a.

10. f^^

=logfa+ a4-V2aa? + a^.

11. (V(a?,^/2ax+a^)iix=

(F(z"a, -Vz^"a^dz,here z=ix-{-a.




Expressions containing a-\-bx"ea?

112. rj^__

^tan-i

^^ + ^.

when6"

"4 oc.

Ja + frx+ caj* V4ac " 6* V4ac " "*

US. =

^log

2"^ + 6-Vy34^^^j^^^

Vft* - 4 ac 2ca! + 6+V6'-4ac

6* " 4 ac.

"6*

114. C ^=

1log Vy + 4ac + 2ca!-i

116. r ^

-=^log(2fla!fe+ 2-^^"+fc" + '"^-

116. CVa+bx + ^dx

*"* 8 c*

117. f^

=J-8m-'.^^-^-"^ Va + 6a;" ex' Vc V6' + 4oc

118. rVa + 6" " cai*da;

= 2^:ilV2rTto3^4.^"i^8m-'^îiJ-.4c 8c* V6" + 4ac

119. r '^^^^

'^ Va + 6a!+ c"*

^Va+ 6"+^__6^1^g(2cr

+ 6 + 2V^Va + 6a;+ caO.2c'i

120. r'"'^

= _

^/a+ bx-cx'

^

b_

^^^_i

2cx-b

^ Va + 6a;" caj*c

2 c*V6'+4ac

Other Aloebraic Exfhessions

121. J'.^^""i(toV^+^(64^)+(a-6)log(Vo+5+V6+").

122- /"\|^"^*=(a-a;)(6+ a;)+(a 6)8m-"^ÎNTEOBAL CALC. " 18




123. C^S^dx = - V(a + x)(b-x) -(a+ b)

sm-"J^^./^0 " X â + 0

124. yyj^-"^dx=^Vl^:r^^Qm''x.

dx

V(a?a){fi x)

125. r-^=^= = 2

8in-xfe^

EXPONENTIAL AND TRIGONOMETRIC EXPRESSIONS

126. Ca'dx=:-^^ 128. Cef^dx^^J loga J a

127. |"'daj"'. 128. |sin a?(to = " cos a?.

130. I cos a;cto = sin x.

131. Jtan a;(la; log sec a; = " log cos x,

132. I cot xdx = log sin x,

133. rsec xdx=i

" " = log (sec; + tan x) = log tan [^-f -VJ cos a?

^

\^2/

134. I cosec xdx = \-4-^

= log (cosec " cot a?) log tan -.

J J smx^^ ^ ^

2

135. Jsec* xdx=i tan a?. 136. I cosec' a?da;= " cot a?.

137. Isec X tan a^da; = sec a;.

138. Icosec a? cot a;da? = " cosec as.

139. fsin'ajda?|-

isin 2".




140. fcoQ^xto = ^H- ~ sin 2 a?.

J 2 4

m Mt /* "- J sin""^ X cos X

.n " l/^"""

j

141. I sm"ajdaj = "

^-=" H I sin""* a? das.

c/n n

%/

142. rcos"a;(to"""'"'^"'"'' + ?^^ fcos-'a^d*.J

'

n n J

143r "^^

__

1 cos a;,

w " 2 r cfa;

J sm"a?~ n " 1

sin**~*ajn " 1

./

sin"~*a;

144 C ^^"

^ sin a;,

n " 2 r dx

*/cos*a? n " 1 COS**"* a? n " 1 J cos**""*aj

145. fcos-a?in"a;cZa?^Q^'-â? sin"^*x

_^m--l Tcos" *ajsin*a;da-.

J w + n m + n%/

148. Icos" a;

sin"x dx

8in"-*ajcos'"+*a; ,n " 1 rr.^""^";r.i"-i".^,*.

= 1 I cos X sm* 'a? oa?.

m-{-n m + nJ

147. f^*^"7

sm*" X cos* a?

_

_1

1 m + n " 2 r (fa?

n " 1 sin^-âjcos""*^ n " 1 J sin'"a/'aos*"*a

148. r ^^

sm" X cos" a;

1 1,

m 4-71 " 2 r dxJ m-{-n-2 r

X cos""* a? m " 1 Jsii" 1 sin*"* a; cos""* x m " l J sin"*"'a cos" x

j^^Q/^cos"*a?da;_ cos**'''* a; m " n

+2 /*cos*"a?da?

%/ sin" a? (n " 1)sin"~* a; n " 1"/ sin""* a

150Cô^'^^dx

_

cos"*~*a? m " l /^cos*""'a?cto

"/ sin"aj (m "

n)sin""*ajm "

nj sii"

151. I sin a; cos" xdx = " " "

-5.n + 1

152. fsin"? cos xdx = 8in;:+^^J w + l

sin" a?






7. }"*, 2a5* -

2jac*,âj^,âj 4a5i

vx

8. logf, log("-l), jlog(a"+ 3),log(tttni?+l),og(a"+ 2aja-2x + 4),


9 g2x

15 (w + n)" 10. " co8 2x, sinSx, tan4x.'

log 2' log(w+n)' 11. secjx, 8m-i2x, Bin-i3x, sin-iwr.

12. tan-i2x, tan-^Sx, 8ec-i2x, sec-i4x, sec-i?.d

Art. 19

2. Jx5, ^\ ^^^, "^^, 26W.v^.11 d

\/2 + l

8. ix"-x2 + 5x, Jx"-2x* + 2x*.

26.S



264 ANSWERS

4. 9t + 16"2 + yf" + c, ort"5(3 60" + c, a^x- {o^x* + fa***- ^x",

sin^ " cos 9.

gsinmx

_

cos3z tan2x_

cot (m + n)x

7.-i-log(a

"x"),ilog(4 + 3"8), - ilog(5-2x*).

no

..

f"in-".,itan-^,tan-Ârt. 20

8. KiK+ a)*"K" +")^log(x

+ a), 2\/ST5,L.,

A(2 + 3x)*5

X + O

-A(3-7x)^.9. sin (x + o),

tan (X + a)," Jtan (4 " 3 x),

"

-cos (o + 6x).

10. 28inf,e^^ -3e-l,--

^

2'' ' '

2sm2x

11. A(4x-3a)(a + x)*,J^(a

+ 6x)*(26x - 3a).

12. log tan-i jp^ gin log x, " n cot- "

n

18.-L

(o + 6")",A

(a + jx)*.^ (a + 6v)^

14. J8in-i2"^.16.

-icosecSa;,isinS^- fsin*^ + isln'^H-ij^sin"^+2sin^, Jtan*0

- Jtan80 + tan^0 + 9tan0.

Art. 21

12.îltan-ix-^

2 2

18. X [(logx)2- 2 logx + 2].

14. sin e (logsin ^ " 1)

16. tan X (logtan x " 1)

Uoga (loga)2^(loga)8r4

9. X tan-i X - pog (1 + x^).17.

-^

e"" (wx - 1).

10. 2 cos X + 2 X sin X " x2 cos x.

11. x^ sin X + 2 X cos x " 2 sin x.

IS. J^hos^-^\



ANSWERS 265

Art. 23

6. log(2a;-6 + 2V"2-5a;). 7. log (7a; + V? V7 "3 + 19).

8. _"_ log (3x2 + 1 + V3V3aj* + 2a-2-l).2V3

9.

ilog^. 21.

-i-log^-^-Â-^4\/3 aj-2 + 2V3

V 2 / 2 22. itan2x + logtanfaj +

j)+ :

11.sin-i-^ "-

loo"-i"--".

y/2

23.

^sec-i0

^^^

4\/5 VE18. 2 log sec 3 05.

14. J_gin-i-X2".

V6 V326. |(x2-y2)!.

16. " tan-i *

VS VS! 27. log(/3+V/32+ 2V3).

4V2 y + V8 Ve o

17.-log sin ox. 29. i tan"! ^-i-?-

18. -log sill(ax + 6). 3Q iogtaii?+ logsin"?.

2

19.-i-tan-i^.

t,

V3 V3 81. - log (a2x+6+a Va2xH2 6x+c).a

20. jtan-i-^.32 v^^H + log (x + Vi^^^H).

Art. 24

8. ^-2xSy-.2xy2+ c. 4. a^

-^- xy^ -

x^ -{"c.

o S **

6. az^ + bxy-^-ay^ + gx + ey^-k.

Page 55

w + n + 2 \3 /

2. Jx" + 4x''-"x* + 16x^, ^J-^s.J + s.^l,



"" ""

it, û '^

"^ ^^.j;u/,-

-/;^HUu fz*~"xi ^^l"v"^nF.

W. W^. ^'^

^

^'"5

6_-| co"za-logco"x),||//^// ^l'/j"/'//'U/ga/ (Wiâ/J

1#,^{C^"K";*f'Kic+ [' 22, tana; " cotx + 21ogtanx.

""' !.^

^'" n f "^/t0)-\-\\(y%(\ + cot" ^).2 2 4

Va'-*f h* a a x^

tr. Min-*'''',

Mill"'''''/^ 80. logBccf^jV



ANSWERS 267

81. log?^5i?. 87. 14sin-i|-.(ix+10)V4^^.cos" $ 2

82. |x* vers-i-+f("+* a)V2 a-". 88.-i-log2V^ seca; + V3

83.

-Vs^"^"^\og(x+y/3^"cfi)."

89. JLlog^-A

84.^

log^-^2\/6 e" + V5

1 /v "

-ir\40. sec-1-^'

86.

-7:ôgtanf|

+

|yVSloga VS

86.

ilogtaii(0+^y41. 2V2

8in-i^V2sm|42. tan-i

^"^^ + ^ when 63 " 4 ac,

V4 ac - 6* V4 ac - ft^

"

1i^,2ax

+ 6-V6^-4(ic^when6a"4ac.

V62-4ac 2ax+ 6 +V62_4ac

43. J-log (2ax + b + 2 Vo Vox^ + 6x + c).Va

44._L siii-i-^^^:l=.

46. Binajcosy + c.

47. 3c8 + 3a;2y + 4i"y2_|.2"" + c

46.-cosxcosy

+ c.

CHAPTER rV

Art. 29

1. (a)"; W4. "" *""" ^'

2. (a) 74 J; (6) f 7. 25|.12- 1"*^-

8. (a)^^;(6)4. 8. f"/I024. w. ^.

6. 28H. 10. 24.^*- ^^''â

Art. 30

6. 81^ ir.

6. (a) ^^^; (6)ifeir;

(c)T"Tr^;

iH"-

7. (a) fTr;(6) 2w; (c)ijîr;(d) V".

8. (a) 47r; (6) iraMog (1+ V2).

9. ^^chî.10. |c*xA 11. ^. 12. i^c^.



268 ANSWERS

Art. 32

4. y = ce"". 6. ya = to"+i + c e. cr = ^.

7.. r* = c8inn^; r=:C8in6; r = c(l"

co69).

- f (-!-)"

Page 76

1._iI"L

!""" 2a2.

^ ^ 11. 4"a".

2-

-i-12.

"a". 8,fl"8. ^/a".

3,^1

!"" ^0og8-2).*"

2 (^"ej* 3.6.7.0* 20. 2ir2(i"."

6. log4-t-^*' *"** 21. Ajiy.

.

7^16. (a) 11^; (")4a2ir2.

9. |a". 16

CHAPTER V

Art. 34

2. log*lli. 7. log(^"^)'."" a5Hlog(x+l)"(aj-4).

8. log("+ 6)2(x-7)8. 8. log[V2^:rf(x+2)].l*- |+6a5-logx"4.

-log(x-2)^(x-l).

"" i^-Sx(x^-^)''(x-p)(x + g)

10. log(a; 3)2(x-2). 16. log^^ '-^ ^.

11. log"-.

^^ x +

log|5|.6. log^^. 12. iog^-2-Â

"^

a;

a;-2 + V3 17. log V"2 +Q

"_

4.

18.TVlog"("-l)"("-2)-"(a;-8)".

^ ^

19.

.^log5.:zV^+-i_logÎi:^.\/2 a;+V2 2\/3 x + V3

20. ilog(a;2 1)- 1 logaj(x2 2)+ Alog(a;2 4).

"""

'^'"vi%iri;7'''^"'--'^"";'-



ANSWERS 269

Art. 35

2. log(a; l)+

-l"

9.alog(a; a)+ -^**

x+1

^

flc+ a 2(a; a)3

8. log(a;-8)?-.

10. logx(x - 1)"-?.

^2 2x4-1 2x + 2 4^x

+ 3

6. log("+ a)"(x+ 6)-" ~ 13. "

L-+ log(x+ l).

X+ 0 X + 2

6. 1 13.-_-^_

+_1_log^"^..3V6-.2-x)" 4(xa-2) Sv^ x-V2

7. log(x-3)3 ?-+ "

? 14.2^-5 2x + l

"^,iog^:zJ.'^^

^^

x-3^2(x-3)2

*

(x-2)2^2(x+l)2^^x

+ 1

^x

+ 1 x + 1 (a;-l)2 "

Art. 36

8. logx + 2 taQ-i X. 6. tan-i x + f^

,^^"

4. log(x+ l)2+ tan-ix.^^ Jlog(3x + 2)- itan-i(x+ 1).

5. ?^4.iiogx+ :5^tan-i-^.8. 31ogx+ " tan-i?^^.3*^3 V2

v^"/2

9. ^-I|!+6x log(x+ 4)"-tan-ix.3 2

10. J-tan-i-^+f" ^^" + f" ^^

11. J_tan-i-^-l.^î+i?.

12. x + hog?^^-V3tan-i-^.6 X* V3

18. log

-^"".4-? tan-i^- Atap-i-^.

'

Vx2 + 2 2 V2 V2

CHAPTER VI

Art. 38

o"

4.^

6.-"Va?^+

q^

"Va3-x" "Vx? + a^ a^x

8X

JVq^ - xg

^(x2-qg)^



270 AN8WEB8

8.

-i\oeit"Ê"R\11.

_y^^^^^lj^/a

+v^TF\

ox

" ".-^"""fVtf^^\ 11. " i

.i^(iM^).10.

--sin-i-.IS.

^"z

Art. 40

4."* + log("*+ l).

\047o54o /

8. A"*(" + ^")*(3""-2a). 9. log(8+ 2 VSTT).

10. 35+1 + 4Vx+ 1 +41og(V"+ 1-1).

11. A(c -

")*(5x + 3 c - 24). 12. I^^"''^

.

^

(2"+3)*

Art. 41

8.

.^Vn^.4.

-îogf^5Z?_=:^^).

V3 \VxM^+V3/

Art. 43

2.

-J-tan-if.^^^^^'^V8.

-i-log(8+ 4x + 2\^V2x2 + 3x + 4).

V2 \v^x

/ V2

4. log(x+l + V2a; + x^)X + V2 X + x"

5. Vx2 + x+l + ilog(x + i + Vx2 + x+l).-

""

-%^-"-(^2^):'"

-^---m



ANSWERS 271

Art. 44

8. V6log(6a;-l+V6V6a;a-2x + 7).

3-i.log(6a;-l

+ 2VSV3aj2-a;+l).V3

4.

8in-i(2LlLl\6.

2V28in-i(i^^y6.

-fV6-3aj-2a;"+

-ii-8in-ifi^"^Vv^ \ \/67/

9. }V3xa_3a; + H--i?-log(6a;-3 + 2V3V3xa-3a; + l).2V3

12. iV6"a-26a; + 34 +-5^1og(5"-13

+ V5V6x2-26x + 84)5V6

+ 13 log(2x-7+V6x'^-26"34\

18.

-log(2+ " + 2Vx" + " + l\

g-114.

2ain-"(^-4^^1og(^+

^J_+^2"-^).

Art. 45

4.-V2ax-ai"+ave"-i-

"" - fx/F^^l^ + ^ sin-i 2â 2 2 a

aaVaa-aC2a"a!" 2a" \ " /

8. - i(3a" + ax - 2 a?)V2 "" - "" + ^ Ters-i?"

". -("'' ^^)^'''--^- 10. J(ai"-2a")x^^T^.

Page 98

1. log (205+ a + 2Va;" + ax). 8. log (2a; 5 + 2 V"" - 6" + 6).

8. ^log[2Vâ; - P+2NVN^^ - Pie + ^].

4. log (a -

a)(2x

- a - 6 + 2V(a; -

a)(a;6)).

5.-i_log(4x"

+ 3 + 2\^V2"6 + 3"" + 1).3V2






CHAPTER n

Art. 12

8. y" = ce"; y" = "". 4. j^ = *(aj" c").

CHAPTER m

Art. 18

5* ir TO + n + l' lOO' 22' 38*

225 40 63fi 9x^ mx* 99x^ 20""

e. *x*, 4"", ""*

-^t^^\ "x^j^*,

-J?_x-?-^\+g g-P

7. }"*, 2"*.--?z, *"i ^xi ^xi 4xi

vx

8. logf, log(* 1), I log(x"+ 3), log(ttw+ 1),log(x"+ 2 xa - 2 X + 4),


9 e2"

15 (m + n)' 10. " C08 2x, sin3x, tan4x.

log 2' log(w+n) 11. secjx, 8lii-i2x, sin-i3x, am-ûv.

18. tan-i2x, tan-i3x, 8ec-i2x, sec-i4x, sec-i?.

Art. 19

^11"i V2 + I

8. Jx"-x2 + 5x, Jx*-2x^ + 2x^.

26.S



264 ANSWERS

4. 9" + 15"2 + iyt"+ c, or

,15(360* + c, a-^- faM + fa*"*-ix",

sin 9 " cos^.

A si^wa;_

cos3g tan2g_

cot (m + n)xm 3

'

2 m+n

'

7.-1-

log (a + 6a;"),ilog(4 + 3t;"), ilog(6-2"*).o

8.

^8iii-i(?,itan-i^,rtan-i^JL* 3 20 4

Art. 20

8. K" +")iK*

+")^log(a; a), 2V^T5,

i-,M^ + Zx)*,

-A(3-7a;)^.9. sin (x + a), tan (a;+ a),

" }tan (4 " 3aj),"

-cos (a + bx),0

10. 28inf,"*w., _8e-i,_.

2'" '

28m""

U. A(4a;-3a)(a + x)*,

j^(a+ te)^(26*-3a).

12. log tan"^ X, sin log x, ~ n cot- -

n

18. i(",+ 62)4,

A("+ 6x)J,

X(a+ 6y)f.

14. jgin-i"iLjZ_i.4

16.-icosec2x,

isin6^-}sin*^H-|8in8^+ V8in"^+28in^, Jtan*^- Jtan80 + tan20 + 9tan0.

Art. 21

6.xsin-ix +

Vn^.12. ?^-"ltan-i "

-^

*

2 2

6. xcot-ix + ilog(l + "^). 13 a;[(logx)2-21ogx + 2].

7 q. [g 1

] .

14. sin^(logsln^-l).Ilog a (loga)2i 15. tanx(logtanx - 1).

8.ax/_^_._i^+_^2_l

16. ^[(logx)2-ilogxi].I log a (loga)2 (loga)8i

4

9. xtan-ix-pog(l+ x*0.

H- ê""(w""-l).10. 2 cos X + 2 X sin X " x2 cos x.

11. x* sin X + 2 X cos x - 2 sin x.

18. ^flogie L.-VOT+1 V TO + iy



ANSWERS 265

Art. 23

6. log(2aj-5 + 2Va;2-6x). 7. log (7 a; + VT V7 x* + 19).

58.-^

log (3x2 + 1 + V3V3x* + 2a;-^ - 1).

2V3

9.

ilog^. 31.

-i-log^-^-Â-^ 4\/3 X-2 + 2V3

10.

6sin-i(-^),or5ver8-i|,;.^,2x +

logtan(x^)-f:11.

Bin-i^^.

l_.x_â

18. 2 log sec 3 X.

14. J_8in-i:^.

4V6 VE

85. ilog(x2 + Vx*-c*).

V5 V3ae. |(x2-y2)l.

16."-tan'

" / 7^

Va V5. 27. log(/3+V/32 + 2V3).

16. JLlogJ^^^:^ 88. J-vers-i^.4V2 y + y/S

Ve ^

17.i log sin ox. 89. 4tan-iîâ 2

18.-logshi(ax

+ 6). jq log tan ? + log sin ^.

19. itan-i2^

V3 VS 81. ilog(a2x+6+aVa^^+26x+c).(Z

80. itan-i"^. ^^ ViaZTi + log (x + Vx^ - 1).

Art. 24

8. ^-2xay-2xy2+ ^+ c. 4. aSx ^ ^- xy^ - x^y + c.

6. axa + 6xj/ + ay2 + â; + ej/+ *.

Page 55

8. Jx" + 4x^-|x*

+ 15x*, 7a*-Hô* + *F-



266 ANSWERS

8.

-^-?^+2a;2-8a;251og(a;+ 2), ^+ "2 + 6" + 51og(" - 2),

24 + 6 log 3.

l+t n-lQ 6 O 4 S

l + " n " 1 1 " n 6 4

5."2 cot 2^, i8in20+ Jcos30, "

-log(a+ 6co8^).

b

6. log(y2-a2), sinx + tanar, iClogx)2,Pog("g + 6)]^

2a

7. 106, i, 1, 4, I, log 2. 8. 2, 1, l-a,".*-"-!"

4 4 2 22 a

9. je". log2,-fl2i2I",

J_i, l(ef_"?).a

10. 4VS,3Jj(_l+3v'3-2V^),

3+Vl5.

12.8in-i^+^,2tan-ivT+li.

13. Jx^-Jx*fcl" V^+T^.

Vl3 3 6-^

14. X sec"! jp _ log(x+ Vz'^ " 1),x cosec-^ x + log(x+ Vx^ " 1),

xcos-ix-Vl-x2, ^sra-ixi(x2+ 2)Vn^.3 9

15. X sin X + cos x, " x^ cos x + 3 x^ sin x + 6 x cos x " 6 sin x,

X tan X + log cos " " " "

16. a"r^3x8

^_6g

6"

1cos xd

-

log cos x),"bog a (loga)2^(loga)8 (loga)*J'^ ^ ^'

cotx(l" logcotx).

17. (x" 2x + 6)sin(x2 2x + 5)+ cos(x22x + 5).

18.^^ t ^'"^^[2log (""+ a") 1]-

a3(x'+ "")[log(x^+ a') 1].6

19. " ^{(logx)2logx + - }" 22. tanx-cotx + 21ogtanx.

o ^4^ 3 9

*

3x

23. ^ 1 log (1+ cot 9)+\og (1 + cot2 6),

L Z 4

24. sin-i^

:"26. 0. 28. 1

sec-i?

-

1 log (x + Vx-" -

o^).

26.

alogtanf|j)+"logsin^.

29. i{log(x+ VxM^)?.

27. sin-i^^, sin-iî^.80.

logsecf^jVV3 3 V2 4/



ANSWERS 267

81. log 5?^. 87. 14 sin-i? - (|x + 10)Vf^^.cos" ^ 2

82. |"*vers-i-+t(a;+4a)V2a-". jg^ 1 logS^^^^^^ '

2V3 secaj+V3

2 21 #"" " V6

Vs Vx + Vb1

1 q*

86. J-logtan(f|]. Vftloga Vft

V2 \2 8/

86.

ilogtan(0+|y41. 2V2sin-i ^V2sm|y

42.

^J^tan-i

^"^^+^, when 62" 4 oc,

V4ac-62 V4ac-6"

1,,^2ax

+ 6-V6^-4^when6a"4ac

V62-4ac 2ax+ 6+V6*-4ac

48.-Llog(2

ox + 6 + 2VoVax2 + 6x + c).

Art. 30

6. 81^ ".

6. (a) 4F"; W A'! WTfff'!

iH"-

7. (a) i"; (6) 2x; (c)^l^r; (d) V-

8. (a) 4ir; (6) ira"log(l+\^).

". ixc"*i*. 10. Ic**ii 11. ^. ". "-"6".



268 ANSWERS

Art. 32

a;+ 2

8. log(" 5)"("-7)".8. log[V2^-[(a!+2)].l*-+6"-log""(*+2"

'"^-^- "--rf^- xe...iog-^-

". log?i^. IS. iog"-2-AX-2+V3 17. logVx" + ^!"!-4.

18. A loga!(a!l)"(x 2)-"(a:3)".

18._!_iog"riyi

J_log?-^

x-l

2V2 a;+V2 2v/3 x+VS

20. Jlog(a;21) Jlogfl;(a;22)+ ^ log(x^ 4).



ANSWERS 269

Art. 35

8. log("+ l)+

-l~.

9. alog(x+ a)+

-^-r7-^-5+1 x + a 2(x + a)^

8. log(x-3) ?". 10. loga;(x-l)"--.

aj " 3 z

3 111

7.ll,_x

+ l4. log V2" + l -^^ 11. " " + ~ log^

2 2aj + l 2x + 2^4*â;

+ 3

6. log("+ a)"(a;+6)-" ^. W. "

i-+ log(a+ 1).

X + O X + 2

" ^ 18.-_-^_ +

-i-log^+^-3V5^2-aj)" 4(x2_2) 8\/2 "-v^

7. log(x-3)" ?L.+_?

14.2x-6 2X+1

_^^x-2^

8. iog-^- +^_.

16. g_+ iog(^-^)l

x + 1 x + 1 ("-l)^ aJ

Art. 36

8. logx + 2taii-ix. 6. tan-i x + J"^" .

4. log(x+ l)"+ tan-ix..,, Jlog(3x + 2)- Jtan-i(aJ 1).

6. ^+ ilogx + ^tan-i-^. 8. 31ogx+ ^tan-i*^.

3 3 V2v^ V^

9. ^-Z^+6x + log(x+ 4)8 - tan-ix.3 2

11. J-tan-"-*" l.^*i"i".

13.

it+ilog?i"5-v/3tan-i4:.

18. log-"^"i-+;tan-i^-Atan-i-*-.

'

8. -

*.



270 AN8WEB8

a 4 Va" + aj*\ --V2cM5-a^

8.--log

-" ^" "^^

. 11. -

ax

8.

-liog(i":^|i"Z)9

-hog/i"îZE?Vw.

^

^ "

*"

a^V

" / "V2cM;-a^

10.--sin-i-.

IS. -"

a X oaxr

Art. 40

8. 8x* +

ilogîlliI'-V8tan-i(2xLtIVj-1 V V3 /

4. a;*+ log("*+ l).

t

m/4-4-^+t+^-4-t+"*+"*-1"i!("'947o54o /

8.fV62(a + 6a;)*(36a;-2a). 9. log(8+ 2VicTr).

10. a; + l + 4Va;+ 1 +41og(Va;+ 1 - 1).

11. A(c -

")*(âj + 3 c - 24). 12. |^^ + ^

.

"

r2"+3)*

Alt. 41

8.-^"^Vrri5.

4. J-logf^^^:-^3

4. J_logf-^^Zzi^).2V2 \\/r=^+V2/

5. V^T6-^logf-^"IziV^\V3 Vvxnps + vS/

Art. 43

2.

_JLtan-if^^î^^^V8. " log(3+ 4a; + 2 V2 V2aj2 + 3aj + 4).

V2 \y/2x

/ V2

4. log(x+l + V2a: + a;2)-a; + "/2" + a;^

6. Vx" + a; + l + ilog(a:+ i + Va;2 + a;+l).-

6.-2/Lz"!

2x3 ."""(va^"i).,

.,^^_.(H=j).



ANSWERS 271

Art. 44

8. V6log(6a;-l+V5V6aja-2x + 7).

3J-log(6a;-l

+ 2V3V3aj2_x+l).V3

4.

Bin-if^y6.

2V2sin-i(i^^^y6.

-fV6-3aj-2a;2+

-ii-8in-ifi^"iV4v^ V V67 /

9. }V3xa-3x + l+-^log(6a;-3 + 2V3V3xa-3a; + l).2V3

12. iV6xa-26x + 34 +^?-log(6x-13

+ V5V6x2--26a + 34)5V5

+ 13 log /2x~7 + V6^2^-26x34\

IS.

-log(2"""2V^"iTl\14.

2sin-i(gL:^)-4V"log(^+

^J+^^^-^).

Art. 45

4.-V2ax-aj"

+ aver8-i* 5. - ?x/5a^^" + ^ sin-i 5.

a 2 2 a

-1 logf^+v^^â8 ^V X Ie

a?Y

^^^ ~ ^^

8. - J(3o" + ax - 2aja)2(ix - x^ + - vers-i?.2 a

". -"''* ^^"^'''^^. 10. \i^-ia^)y/WT^.

Page 98

1. log (2x + a + 2 Vx2 + ax). 8. log (2x r- 5 + 2 Vx^ - 5x + 6).

8. ^log[2mx -P+2NVN^^- PbB + ^1.

iv

4. log(x -

a)(2x

- a - 6 + 2 V(x -

a)(x

- 6)).

6.-i-

log (4x8+ 3 + 2V2 V2x6 + 3x" + 1).3^



272 AN8WEB8

e.-i-sin-iV3(x

+ l). 7."log [V5(x + 1)+ V6a;2 + 10" - 27].

8. --JLsin-i!?*?. 9. f +

JLsin-irx-n+

A "-l,

Vm " 17 17 17V2x-a^

10. ilog(2x + V4gg - 9)+

-^sin-i^^^^\

11. ~ 2 Vx2 + 2x + 4 + 6 log (a;+ 1 + Va;2+ 2a; + 4).

12. - J (2a; + llV(a; - 2)(3 -

aj)JgS^

sin-i (2x - 5)

18.

i ("-

4)Vx2-T+

}log(x+ V"2 _

1).

V2*

V aj + i y \ 2x+3 y

16. log(x + 2+Vx^ + 4x + 6)+ ^log(^^^'+^^^+^-

18. tan-iVa? - 3.

19. v/;^M^ + 21og(* + V;?+T)-Alogf3^:*"^f^+l)2 \ x+ 1 /

2^1-x 4V2 W + 3 /

21.^x*

+ a62a;a+ LV"+lîS b^ + 2a^^) +

;^log

Vb^ + a^^ + ax).2

4a

22 xC3o2-2xg)

3a*(a2-x2)*

28.-iL "/a2

-

x-"(8x*lOa^x^ + 16a*)+^8in-i5.8 16 a

24. - Va2 - x2. 26. -

yVv'a"- x2 (3x* + 4 a^â + So*).

26.

_llog("+^JHE^).27.

V5^3^"-alog("^t28. ^'VS^"^^ (2 x2 -

a2)+ ^sin-i^.8 8 a

29.-?-

Va2 - x2 (8x* - 2 a^x* - 3a*)+ ^ sin-i

?"

80. zpv^l^^511

_lwr""2^"Z^^x

81.

_llog(""^^

+

"?).82. ^ Vx2 " a2(2aj2" a^) ^ log (x + Vx^ " o"). 88.

^^

^"

8 8rtWxa " a^



ANSWERS 278

84. IVxa "a^{2x^"b "")+ ^ log (x + Vx^ db o-*) S6. Vsc^To^.o 8

86. -\/2aa;-a;2 + avers-i^. 87.-Jî^^^ZZ.ox

3g,2x"

+ 6a(x + 3a)^^^__â+6"!ver8-ig.

6 2 a

89. i^V2ax-x^ + ^%er8-i^. 40. "(3f 6ag) 3^^.,x

2 2 a 8a*("^ + "*)^ 8a6 a

41. ^ +

_l_tan-i4

48.

x + 1

x-1

4a*(x* + a*) 4a" a^

'

3(a"-x")

48. ^^^ + J__.tan-if?l:=iV4- + Hog4(x2-2x + 3)^4V2 W2 / 2 (1 -

x^)

^ ^ ^

46. Vl+x+x-^-ilog(2x+l+2Vl+x+x-^)-logf^""^+\ x+ 1 /

46. - tan-i x + 2 \/2 tan-i a/5 V3 tan-i

-v/|.2 '3

47 nin-12 a'^*' ("'+ "')(a' + fe" "") 4. :f"i

(a"-6")(a"+ 6"-a!")'

4

61.2x + 6

^^^2 ^_ 4a. ^_ 3 ^ |iog(2g + 1 +\/4x-^ + 4x + 3).

62. 5-=^Vx2 + 2xH-3. 63.

--^V- 8 + 12x-9x2-||sin-i(3x-2).

2 lo

54. "(2to + m)Vte.4.^4.^ + ^^-4Zn^.^,W^jx + m_\

4^ 8iV-Z V\/m2-4in/

551

w r(" + V^^^="^)' + (3 - 2 V2) q2 .

2aV2L(x

+ Vx'*-o'^)*^ (3 + 2V2)a2J

gg2V3xg-2x + l

2x-l

CHAPTER VII

Art. 46

6. (o) sin X " 8in" x + ^ sin*" " | sin"'x ;

(6) " cos X + J cos'x " i cos* X ;

(c) " cos X + cos8 X " I cos* X + ^ cos^ X.



274 ANSWERS

(6)-

Jsin6xco8x+

jfsin*x"to,see

(a)];

(c)-?i5!"."2i^-rsin"xda;,

see(6)].7 8

"/

8

(6) Jsinajcos^x+ ircos^xdc,see(a)].

(c) j8inajcos7x+|fcos6x(to,see(6)].

8. (a)-li2|^

+ ?f-rV-"seeEx. 4).^ ^

4 8in* 05 4 J sm" jk

(6) i secx tanx + logVsec a; + tanac

(d)li5M.+ 3C_^, [see(6)]."

-' 4cos"ie 4J co8""

"" ^ ^'

U)liiH^

+ lf-^. [8ee(c)].'-'5 cos's; 5 J C08*ac

"" "^ '""

(/")_1.52?"_"cotx

+ c.^"'^

3sin"x^

(J,)_i^

+ |f^, [8ee(/)]."^

Ssin^x 5^ sm*x

Art. 48

4. (a) Jtan8x4-tanx + c. (c) - icot^x- f cot'x - cotx.

(6) - Jcotsx-cotx. ("i)tan8|3tan|+c.6. (a) Jtanxsecx + ^logtanfîy

(6) Jtanxsec8x + I|tanxsecxlogtanfj+ |H.

(c)-20t^^!"""+ilogtanf. x\

(^) " jcotxcosec^x " { (cotxcosecx " logtan- !"

(6) ftanfxsecfx +

flogtan^j+iyArt. 49

2. (o) itan"x-tanx + a;. ("0 - i cot^ x - log sin x.

(6) Jtan*x-itan2x + logsecx. (e) - Icot^x+ cotx + x.

(c)itan6x-Jtan8x + tanx-x. (/) - J cot* x + i cot2 x + log sin x.



ANSWERS 276

Art. 50

2. I sinâ; "

T"j-sin3 a; 4- ^y sin

'"x. 3. " ^ cos* a; + A cos

*x.

4. - 2 Vcos a; (1 " I cos^ x + i cos* "). 6. |sin'x(l " ^sin^x + i sin*x).

6. J tanfi X + itan* x + e. 7. " } cot^ ^ " i cot^ a; + c.

Art. 51

2 cosa;/8inâ; sin^x sinx\ .x - cosg

xqp^pgyN

^^

2^3 12 8 J'îe"

2sinx^^

2'

POS X ^

3. tanx " 2 cotx " 4 cot*x. 6. ^" cosx " | log tan*.^

2sin2x

^ ^

2

Art. 52

2. itan2x + 21ogtanx-icot2x.^ _

cosec^x

^ ^ ^^^^^.^ __ ^ ^^^^^^^

y

3.T^^tan-Vx

+ ftan^x. $. Jtan^x + ^tan^x.

4. " J cot" a; " I cot* x. 7. 4 sec* x (^gsec^ x "

^^rsec x + J).

Art. 53

2. i/^-sin2x+^^y 8. |x + isin2x +uVsin4x.

4.Y^^ (5x " 4 sin 2x + J sin'2 x +

-}in 4

x).

6.3iy(5x

+ 4sin2x " Jsin82x+ |sin4x).

6.

_sin^J^^^^_sm4x.7.

î^/sx-

sin4x +

"""^'

(3tan?-2)

4.

itan-i(2anfV1.

-Ltan-i? ^ ^^

/- K 1 1""tan X + 3

3

_i_log2tanx

+ 3-v^.5. ilog^-"

^.2V5 2tanx + 3 +

v^

3.

ltan-ifitan?V6. Jlog^^ÎL^V 2/

^ ^

tanx + 2

Art. 56

1. i6'(sinx-cosx).4- Ac-3"(2sin2x-3co8 2x).

o 1 ,/"

.. \

" 5. " ie-*(sinx + cosx).2. J

e'(sm

X + cos

x).

2 v i^ y

" V A ,

^,/i,2sin 2x + cos2x\

3-TV"^'(3sin3x

+ 2cos3x).""i"'^l ^

)"

INTEGRAL CALC. 19

48 16 64 """V 8

Art. 55



276

I _

cos 8 a; cos 2 x

16 4

" sin 1 1 X. sin 3 X

^*"l2~'^~6~*

CHAPTER VIII

Art. 59

4. 240. 5. 80.

6. Tlie double infinity of straight lines y = cix + Cj, in which ci, cj are

arbitrary constants.

7. 3y = 2x(x2-l). 8. i(n^ -

mfi)(d c)(6-

a). 9.ir(/3 o).

Art. 69

8.Tab. 4. 5ir2a".

Art. 70

8. 2irâ^b. 6. jira2^. 7. (t - ^)a'^h.

4. jm. 6. 4} cubic feet. 8.irVpqh^.



ANSWERS 277

Art. 71

4a2a

= VSSTT^^ + a log ^^L+^+a.Va

8 = 2.29558 a.

5. s = 4

a(cos

^" cos

^J;

length of a complete arch = 8 a.

6. 8 = 6a.

Art. 72

3. 8a. 4. "r"?,vT+^-îVnMr2 + log??-"^4:"^l-

vr2(a+Va2 + ri2)J"

7. 2a/V6-2-V31og^ + "^ 1.

I v^(2+V3))

=

atan^8ecalogtan(^j^;" =

2a(sec+ logtanf

rV. s

Art. 73

1. s = r0. 2. s = 4a(l cos^"

3. (1) " = 4a(l "

COS0); (2)8 = 4asin0.

4. (1) "=ptan0sec0 +plogtan (^+ jj;

(2) 8=p tan (0+1]sec^0+|Wi)logtan|+^]pV2-i)log2p\/2.5. 9* = 4a(sec80 " 1). 8. s = clog sec 0. "

6. , =

"logtan(|l). ^ . = ^8in.0.2

7. " = a(e"* 1).



278 ANSWERS

Art. 74

8. 2xb /"+ "

-cos-^^Vor 3. }iraJ(a+ x)*'

V Va2-

62 ay, ,.

2ira6fVl-6"+?^5::i^V^. "/

V e r 5. V'"'-

in which e is the eccentricity.^'

*"(*" 2)a^.

7. ^^Tra^

10. ^-1",or 32.704".

11. in".

5. fa.

6. I a.

Page 164

1. fro*.

a. Volume = J7ra25;surface = 2ira2 + " log?-"-^,in which e is tlic

eccentricity.^ "" ^

8.

x{(x+|)v5rnr2--fiog^^+

^+^^^^^^"+^}.

4. 4ir2a". 6. 7r6\/a2 + 6i. 6. 4ir2a6.

7.

faft^cota.8.

ia%.9. Volume = Y- a^ ; surface = 8 a'^. 10. | ira*.

.

11.y*5 ira^.

12. Volume =

2"a"(^^'""g^"^'''-ec08.);surface = 4

ira^(sin" a cos

a).

13. Volume = irâ' ; surface = V ^^^2. 14. Surface = ^ ira^.

16. Volume =

^^8^" -

?V surface = 8

)ra2(rj).

le: 1^(10-3ir).

17.3J5ir(3a2

4a6 + 8 62)A.6



ANSWERS 279

18. 2ay/S. 19. 8=Vb^-\- y' + b log^^'' + ig-ZlJ,here 6=

--^.y log a

22. logVS. 28. ^. 94. ?^. 96. ^[W + 4)i-8].

CHAPTER X

Art. 79

5. The density at a pohit three fourths of the distance from the vertex to

the base, namely, ^kh.

6. x = ih, 7. 2c = }^; x = }A.

8. x = y = }a. 9. x = 6. 11.

x=|^y = 0,

19. Mass = f *a", if density = jfcistance.

Mean density =

.4244max. density.

Center of mass is at y = 0, S =

^^ ra =.680 a.

18. ^ = 0,x =25"in". 14. a8 =

i",F=16.

o a OT OT

15. (a)x = ih,y = 0;

(b)x = ih, y = ik, in which ji;s the ordinate corresponding to as = ^.

17. x = y = JJf.^. 18. x = y = ia. 19. x = 4a,y = 0.

90. X ="

I a, 1/= 0. 94. At a point distant f a from the base.

91. x = }",y = 0. 96. ic =

-|a,y= 0.

Art. 80

(Inthe answers M denotes mass)

4. If a is the radius, I=iMa*j k =

-^.

y/2

5. /=:"M"i"J^. 6. A;=

-^.7. ^.

12 V6 20

8. (o)/=i3f62; (6)/ = J3fa2; (c) / = Jif (a2+ 6^).

9./=3f^.

10. /=i3f(62 + c"). 11. /=J3fa2. 19. I=iMa\3

18. /= iâ" = i 3fa2, since 3f = f jfca*by Ex. 12, Art. 79.

15

14. I=Mm., u =^.

/=m,W^ + f\ or 3f(^%n.



280 ANSWERS

CHAPTER XI

Art. 82

2 62.4.9 2.4.6. 13

4 ?+l a;^.1 ' 3 a"

,1 " 3 . 6 x^

*

7 2* 6 2.4 11 2.4.6 16*

6. }x*(l-ia;2-^VaJ*-ifVa^ +^-)

+ c.

^' * \,m^ q m-Hn^ 1 . 2 . g2 w + 2"+ j^^'

an^/:j:rzfi

.

1 sin^x .

1.3 sin* x,

\,^

". logx + x + ^^ +

3-^^+...+c.

3|3^6[5 7[7^^

Art. 83

^23^2 4 5^ 2-4...2n 2 n + 1

6.

log(l+x)=f-||-^+....

log(l-x) =

-^-^-^-^.^ ^12 3 4

^V^

2 3 2-4 5 2.4.6 7

CHAPTER XIII

Art. 97

Art. 98

2. yVl -x-^ + xVl -y^= c. 8. tany = c(l

-

e*)".



AN 8 WEBS 281

Art. 99

. 2. a;jfa c2(a; 2y). Z. y = c^.

Art. 100

3

8. xV + 4jB8y-4xy8 + y"-xe"' + c2"y + a5* = c.

Art. 101

a. 2alog" + alogy-y = c- 8. "a-i^-l = "a5.

4. xV + my2 = cx*.

Art. 102

2. y = (x+ c)"-". 4. y(x2+ 1)2= tan-la; + c.

1

8. y = x2(l+ C"^). 5. ""y = 0x4-0.

Art 103

a. 7y"* = ex*- 3x". 8. y^ =

c(l-

x^)*-i^-

4. 60 y"(x+ 1)2= 10 x^ + 24 x^ + 15^* + c.

Art. 104

2. 343(y+ c)8= 27axL 8. (y -

c)(f/ x2.- c)(xy+ C2^+ 1)=0.

Art. 105

a. log(i)-x)=

-^+c, with the given relation. 8. x=logp2+6p+c.p" X

Art. 106

1.y=c-alog(i)-l),x=c+alog^.

.

y-c=V^^-U,n-^^-

Art. 107

8. j/= cx + sin-ic. 4. j/=

cx+^.6. y^^cx^ + l+c.

Art. 108

8. The catenary j/= ^(e"e~")r

^= cosh-i-.

4. The envelope of the family of lines y = ex +

-^,namely the parabola

(x-2/)2-2a(x + t/)+a2= 0.

^





ANSWERS 283

Art. 114

-4t

4. y = cie-2* + e'(c2os V3 x + Cs sin \/3 a).

6. y = cie-' + C2e -2* + csc^*. 6. y = Ci6-*" + C2C"* + cje*.

7. y = ci6*" + C2e-*" + Cs sin(ax + o).

Art. 115

2. y = ci + e-*(C2 cjx). 3. y =

e-*(ci+ C2X + csx^) cie**.

Art. 116

8. y = Cix"i + C2X-2. 4. y = x\ci + C2 logx).

6. y = x2[ci+ C2logx + C8(logx)"].

Page 227

2. y2 = 2xy " + x". 12. y-"+i=ce(" i)"n*+2sinx+-^.dx n"1

8. tanx tan y = k^, 13. JL= ^^ + 1 + cc*'.

iV.y

4. x" + y^ = 2anan-i|c. ,^^ ^^^^ ^^^^^. ^^^^,^^,

6. xy{x"

y)=c.

6. ax2 + 6xy + cy2 + grx + ey = ife.l^- 2y=cx2 + ^.

c

7. x = cye^.16. y2_.2cx + c2.

8. x2 + " = c.

y 18. (cix+ C3)2+ a = c,y".

X 1""

^^Y^â^^^' ^^' y = "Ji + "^2" + cse** + C46-

10. y = tan X - 1 + c"-t"".^0. y = Ca - sin-i cie-".

11. y = ax-\-cx Vl - x^. 21. y = Cie-* + C2 + |e'.

X

222. y = Ci"-* + "2(

2 cos " X + Cs sin -

23. y = c sin (nx +o)(ax H- 6).

24. y =

"'(ci+ C2X)sin x + e*(C8+ C4X)cosx.

25. y = ci + C2X + e*(c8+ C4X).

56. y =

x(cicoslog X + C2 sin log

x) + CsX"!.

27. y = (ci+ C2 log x) sin log x + (cs+ C4 log x) cos logx.

28. J?/y =

|^?|!-|J4-CiX

C2..

29. j"/y =

-7"-v-+cix3-fC2X2 + C8X + C4.





INDEX

[The numbers refer topages.]

Algebraic tran'sformatious, 103, 100, 107,

.

108, 113.

Amsler, 188.

Amsler's planimeter, 189.

Angles, use of multiple, 114.

Anti-derivative (anti-dififerential),, 5,

7, 11, 12, 14, 21, 25.

Applications to mechanics, 167.

Approximate integration, 177-189.

rules for, 181-188, 2:i6.

Areas, change of variable, 141, 142.

derivation of integration formulae

for, 9, 27.

oblique axes, 140.

polar coordinates, double integra-ion,

139.

polar coordinates, single integra-ion,

135.

precautions in finding, 63.

rectangular coordinates, 58.

rectangular coordinates, double in-egration,

126.

rules for approximate determination

of, 181-188, 235-237.

surfaces of revolution, 152-156.

surfaces z"/(a;,y),

156-160.

AnziHsrjr aquaUon, 223, 224.

Bernouilli, James and John, 2.

Bertrand, 237.

Boussinesqt 186.

Cajori,2.

Gardioid, area, 138.

center of mass, 173.

intrinsic equation, 151.

length, 149.

orthogonal trajectories,16.

surface of revolution, 156.

Carpenter, 189.

Catenary, area, 76.intrinsic equation, 160.

length, 147.

surface of revolution, 156.

volume of revolution, 77.

Center of mass, 168.

Change of variable, 41.

Circle, area, 61, 139, 140, 141.

evolute of, 166.


length, 146, 149.


Cissoid, center of mass, 173.

length, 149.


Clairaut, 213.

equation of, 213.

Complementary function, 222.

Cone, center of mass, 172.

moment of inertia, 175.

surface, 164.

volume, 71, 73, 143.

volume of frustum, 77.

Conoid, volume, 143.

Constant of integration, 22.

geometrical meaning of, 23.

two kinds, 25.

Convergent, 178.

Cotes, 236.

Curves, areas of, oblique axes, 140.

areas of, polar coordinates, 135-140.

areas of, rectangular coordinates, 58.

derived, 29.

derivation of equations of, 26, 26, 75,

134, 214-217.

integral, 3.3, 190-200, 240-245.

intrinsic equations of, 149.

quadrature of, 58.

Cycloid, area, 141.


285



286 INDEX

Cycloid, length, 147.

note on length, 144.


volumes and surfaces of revolution,

166.

Cylinder, 143.

moment of inertia, 176.

Density, 167.

Derived curves, 29.

Derivation of equations of carves, 26,

26, 76, 134, 214-217.

Derivation of fundamental formulse, 48.

Differential equations, see Equation.

Differential, integration of total, 62.

Differentiation under integration sign,

190.

Durand, 186, 187, 190, 237.

Equations, homogeneous in x, y, 206.

linear, constant coefficients,223.

linear, homogeneous, 226.

linear of firstorder, 208.

linear of nth order, 222.

linear, properties of, 222.

of order higher than first,218-228.

reducible to linear form, 209.

resolvable into component equations,

210.

solutions, general, 202.

solutions, particular, 202.

solvable for x, 212.

solvable for y, 211.

variables easily separable, 206.

Evolute, of circle, length, 166.

of parabola, length, 165.

Expansion of functions in series, 179.

Exponential functions, 117.

Figures of curves, 246-248.

Fisher, Irving, 30.

Folium of Descartes, area, 138.

Formulae, areas, 9, 27, 137, i:"9, 141.

lengths, 145, 148.

of approximate integration, 183, 185,

186, 187, 235-237.

of integration,fundamental, 37,47, 48.

of integration, table of, 249-262.

of integration, universal, 39, 44.

of reduction, 46, 93, 94, 95, 101, 102,

106,110,231-234.

surfaces, 154, 155, 157, 168.

volumes, 70, 130, 143.

Fourier, 9.

Fractions, rational, 78-83,.229-231.

Functions, irrational, 84-99.

trigonometric and exponential, 100-118.

Geometrical applications, 68-77, 126-166,

214^217.

meaning of constant of integration,

23.

principle, 7.

representation of an integral, 14.

Graphical representation of a definite

integral, 73.

Gray, 189.

Gregory, 180.

Hele Shaw, 189.

Henrici's report on planimeter, 189.

Hermann, 188.

Hyperbola, area, 68.


related volumes, 77.

Hyperbolic spiral, area, 138.

length, 149.

Hypocycloid, center of mass, 172, 173.


length, 147.

surface, 156.


Integrable form, 37.

Integral, complete, 203.

Integral curves, 33, 190-200.

applications, 195-193.

applications to mechanics, 196, 240-

242.

applications to engineering and elec-ricity,

242-244.

determination of, 198.

relations, analytical, 192.

relations, geometrical, 194.

relations, mechanical, 195.

Integral, definite, 8.

definite, evaluation by me.asuring

areas, 181.

definite, geometrical representation,

14.

definite, graphical representation, 73.

definite, limits, 9.

definite,

precautions

in finding, 67.

definite, properties, 15.

definite, relation to indefinite, 24.



INDEX 287

Integral, elliptic, 147, 180.

general, 23.

indefinite, 22.

indefinite, directions for finding, 54.

multiple, 120.

name, 1, 21.

particular, 23, 204, 222.

Integrals, derivation of, 48.

fundamental, 36-38, 47.

table of, 249-262.

Integraph, 198, 200.

theory of, 244.

Integrating factors, 207.

Integration, aided by changing variable,

41, 141.

approximate, 177-189, 235-237.

by parts, 44, 46, 100, 231, 233.

constants, 22, 202.

definition, 1, 18, 21.

derivation of reduction formulae, 231,

233:

fundamental formulae, 37, 47.

fundamental rules and methods, 36.

in series, 177-179.

mechanical, 188, 189, 200.

of a total differential, 52.

precautions, 63, 67.

sign, 2, 21.

signs in successive integration, 125.

successive, one variable, 119-123.

successive, two variables, 123-125.

universal formulae, 39, 40, 44.uses of, 1.

Intrinsic equation of a curve, 149.

Irrational functions, 84-99.

Jevons, 30.

Lamb, 237.

Laurent, 180.

Legendre, 180.

Leibniz, 2, 59, 144, 180.

Lemniscate, area, 137.

Lengths of curves, polar coordinates, 147.

rectangular coordinates, 144.

Limits of a definite integral, 9.

Logarithmic curve, area, 76.

length, 165.

Logarithmic spiral, area, 137.

length, 149.

Markoff, 186.

Mass, 167.

Mass, center of, 168, 169.

Mean value, 17, 160-164.

definition, 163.

Mechanical integration, 188-189, 200.

Multiple angles used, 114.

integral, 120.

Neil, 144.

Newton, 2, 69, 73, 144, 180, 236.

Oliver, 4.

Orthogonal trajectories,14-217.

Parabola, area, 5, 14, 59, 68, 76, 126, 138.

center of mass, 173.

derivation of equation, 26.

intrinsic equation, 151j152.

length, 146, 149.

length of e volute, 165.

orthogonal trajectories,16, 217.

semicubical, area, 68.

semicubical, intrinsic equation, 152.

semicubical, length, 144, 147.

surfaces of revolution, 155, 164.

volumv of revolution, 71, 77.

Parabolic rule, 184, 186, 236.

Paraboloid, center of mass, 173.

volume, 131, 143.

Pascal, 59.

Planimeter described, 188, 189.

theory of, 237.

Pyramid, volume, 143.

Quadrature, 58, 73.

Range, 161, 163.

Rational fractions, 78-83.

decomposition of, 229-231.

Reciprocal substitution, 84.

Reduction formulae, 46, 93-95, 101, 102,

106, 110, 231-233.

Roberval, 58.

Signs of integration, 2, 21, 125.

Simpson, Thomas, 184.

one-third rule, 182, 184, 186.

three-eighths rule, 236.

Slope of a curve, 25.

Solids of revolution, surfaces, 152-156.

volumes, 69.

Solutions, general, 202.Sphere, surface, 155, 158.

volume, 131, 132, 133, 143, 164.





BY THE SAME AUTHOR

DIFFERENTIAL EQUATIONS

AN INTRODUCTORY COURSE IN DIFFERENTIAL EQUATIONS FOR

STUDENTS IN CLASSICAL AND ENGINEERING COLLEGES

334 pages. $ 1.90

'The aim of this work is to give a brief exposition of some of the devices

employed in solving differential equations. The book presupposes only a knowl-dge

of the fundamental formulae of integration, and may be described as a

chapter supplementary to the elementary works on the integral calculus/" Extract from PrefacfS.

In use as a text-book in Johns Hopkins University, Baltimore, Md. ; Van-

derbilt University, Nashville, Tenn. ; University of Missouri, Columbia, Mo.;

Purdue University, La Fayette, Ind. ; Wesleyan University, Middletown, Conn. ;

University of Toronto, Canada; Cornell University, Ithaca, N. Y. ; University jDfDenver, Denver, Colo. ; Armour Inst., Chicago, 111.; and other leading insiitutions.

'We commend the book as providing an excellent introductory course in

Differential Equations.' " American Mathematical Monthly^June, 1897.

*The book seems to be an excellent practical introduction to differential

equations, containing a well-proportionedand suitable treatment of most of the

topics which the student needs in his first course in thesubject,and of these only,

a good variety of exercises, and enough historical and bibliographical notes to

suggest further reading.'" Bulletin of the Americati Mathematical Society,

March, ISJ^.

' The work, which we have read with considerable interest, assumes in the

reader little more than a knowledge of the fundamental formulae of integration,

and brings in many practical applications well adapted for the class of studentsfor whom it is intended. The rigorous proofs of many of the theorems are

relegated to the appendix to be read when some familiarity with the subjecthasbeen acquired. An interesting feature are the numerous historical and biographi-al

notes scattered throughout the text, and there are full indexes of names and

of subjects which add to its utility.' " The London, Edinburgh, and Dublin

Philosophical Magazine, October, 1897.

' Mr. Murray's book is adapted to provide for students that knowledge of the

subjectof differential equations which they are likely to want in applications of

mathematics to physics, and in the general courses in arts and science in ''clas-ical"

colleges. The author is chiefly occupied with giving expositions of the

devices usually employed in the solution of the simple differential equations

which such students meet with, and he will be found a safe guide in these mat-ers.

He follows the plan,which most recommends itself to teachers, of omitting

theoretical considerations, or postponing them until the student has had practicein carrying out the processes with which he must be acquainted before the theory

can be understood.'" iVa aire, 10 Feb., 1898.

*The subjectis explained in a simple and intelligible way ; and a very large

number of references and interesting historical notes are given, which cannot

fail to excite the reader's ambition to learn more of the subject. Another feature

is that in some cases a theorem is only stated, while the proof is postponed to a

note at the end of the book. This plan, if not employed too often, seems a goodone ; and in the case of difficulttheorems it would be well if it were more com-monly

adopted.'" The Educational Times, 1 Nov., 1897.

LONGMANS, GREEN " CO., Publishers

91 " 93 Fifth Avenue, New York

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An Elementary Course in the Integral Calculus

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