CalculusCalculus

Chapter 8Chapter 8Principles of Integral EvaluationPrinciples of Integral Evaluation

By: Rhett ChienBy: Rhett ChienEdited: Anna LevinaEdited: Anna Levina

8.1 An Overview of Integration 8.1 An Overview of Integration methodsmethods• ∫∫dx = x + Cdx = x + C• ∫∫xxrr dx = ((x dx = ((xr+1r+1)/(r+1)) + C and x cannot = 1)/(r+1)) + C and x cannot = 1• ∫∫eex x dx = edx = exx + C + C• ∫∫r dx = r ∫ dx = rx + Cr dx = r ∫ dx = rx + C• ∫∫dx/x = ln|x| + Cdx/x = ln|x| + C• ∫∫sin x dx = -cos x + Csin x dx = -cos x + C• ∫∫cos x dx = sin x + Ccos x dx = sin x + C• ∫∫secsec22x dx = tan x + Cx dx = tan x + C• ∫∫csccsc22x dx = -cot x + Cx dx = -cot x + C• ∫∫sec x tan x = sec x + Csec x tan x = sec x + C• ∫∫csc x cot x dx = -csc x + Ccsc x cot x dx = -csc x + C

8.2 Integration by Parts8.2 Integration by Parts

• Method of integration based on the producMethod of integration based on the product rule for differentiationt rule for differentiation

• ∫∫f(x)g(x) dx = f(x)G(x) - ∫f’(x)G(x) dxf(x)g(x) dx = f(x)G(x) - ∫f’(x)G(x) dx can also be written as:can also be written as:

∫∫u dv = uv - ∫v duu dv = uv - ∫v duu=f(x) du=f’(x) dxu=f(x) du=f’(x) dxv=G(x) dv=g(x) dxv=G(x) dv=g(x) dx

Integration by Definite PartsIntegration by Definite Parts

b b b b b b

•∫∫u dv = uv| - ∫v duu dv = uv| - ∫v du a a aa a a

Examples~Examples~∫∫xx22eexx dx dxu= xu= x22 dv= e dv= exx dx dxdu= 2x dx v= edu= 2x dx v= exx

∫∫xx22eexx dx = x dx = x22eex x - ∫e- ∫ex x 2x dx2x dxu= 2x dv= eu= 2x dv= exxddxx

du= 2dx v= edu= 2dx v= exx

∫∫xx22eexx dx = x dx = x22eex x - 2xe- 2xexx - ∫2 e - ∫2 ex x dxdx = x= x22eex x - 2xe- 2xexx - 2 e - 2 ex x + C+ C

22∫∫xx22lnx dxlnx dx 11u= lnx dv= xu= lnx dv= x22 dx dx du= dx/x v= (1/3)(x)du= dx/x v= (1/3)(x)33

2 2 22 2 2∫∫xx22lnx dx= (1/3)(x)lnx dx= (1/3)(x)33lnx| - ∫ ((1/3)(x)lnx| - ∫ ((1/3)(x)33)) (( 1/x)dx1/x)dx1 1 1 1 1 1 22∫∫xx22lnx dx= (8/3)ln2 – (7/9)lnx dx= (8/3)ln2 – (7/9)11

∫∫Calvin dx = Hobbes - ∫Calvin dxCalvin dx = Hobbes - ∫Calvin dx2 ∫Calvin dx = Hobbes2 ∫Calvin dx = Hobbes∫∫Calvin dx = Hobbes/2Calvin dx = Hobbes/2

8.3 Trigonometric Integrals8.3 Trigonometric Integrals

• n is + and oddn is + and odd∫∫sinsinnnx dxx dx= ∫sinx*sin= ∫sinx*sinn-1n-1x dxx dxUse sinUse sin22x=1-cosx=1-cos22xxPythagorean identitiesPythagorean identities

• n is + and evenn is + and even∫ ∫ sinsinnnx dxx dx Use half angle formulasUse half angle formulasSinSin22x= (1-cos2x)/2x= (1-cos2x)/2CosCos22x= (1+cos2x)/2x= (1+cos2x)/2

• m and n are evenm and n are even∫ ∫ sinsinmmx cosx cosnnx dxx dxUse half angle formulasUse half angle formulas

• m is odd and both +m is odd and both +∫ ∫ sinsinmmx cosx cosnnx dxx dx= ∫ sin= ∫ sinm-1m-1x cosx cosnnx sinx dxx sinx dxsinsin22x=1-cosx=1-cos22xxU-substitutionU-substitution

Examples~Examples~

∫∫sinsin44x dxx dx= = ∫ ((1-cos2x)/2)∫ ((1-cos2x)/2)2 2 dxdx=(1/4) ∫1-2cos2x + cos=(1/4) ∫1-2cos2x + cos222x dx2x dx=(1/4) ∫1-2cos2x + ((1+cos4x)/2) dx=(1/4) ∫1-2cos2x + ((1+cos4x)/2) dx=(1/4) (x – sin2x + (1/2)x + (1/8)sin4x) =(1/4) (x – sin2x + (1/2)x + (1/8)sin4x)

+ C+ C

∫∫sinsin55x dxx dx=∫ sin=∫ sin44x*sinxx*sinx=∫(1- cos=∫(1- cos22x)x)2 2 sinx dxsinx dx=∫sinx*(1 - 2cos=∫sinx*(1 - 2cos22x + cosx + cos44x) dxx) dxLet u = cosxLet u = cosx du = -sinx dxdu = -sinx dx= -∫1 – 2u= -∫1 – 2u22 + u + u4 4 dudu= -cosx + (2/3)cos= -cosx + (2/3)cos33x – (1/5)cosx – (1/5)cos55x + x +

CC

8.4 Trigonometric 8.4 Trigonometric SubstitutionsSubstitutionsEvaluating integrals containing radicals by making substitutions invEvaluating integrals containing radicals by making substitutions inv

olving trigonometric functionsolving trigonometric functions

• x=a*sinx=a*sinθθ x=a*tan x=a*tanθθ x=a*sec x=a*secθθ

Examples~Examples~

∫∫dx/(xdx/(x22√x√x22-4)-4)

√ √xx22-4 = 2 tan -4 = 2 tan θθ•

=∫(2sec =∫(2sec θθ2tan 2tan θθ) / (2 sec ) / (2 sec θθ))22(2tan (2tan θθ) d ) d θθ=∫(1/4)cos =∫(1/4)cos θθ d d θθ=(1/4)sin =(1/4)sin θθ + C + Csin sin θθ = (√x = (√x22-4)/x-4)/x(1/4) [(√x(1/4) [(√x22-4)/x] + C-4)/x] + C

8.5 Integrating Rational 8.5 Integrating Rational Functions by Partial FractionsFunctions by Partial FractionsDecompose a rational function into a sum of simple rational functions that Decompose a rational function into a sum of simple rational functions that

can be integratedcan be integratedProper Rational function = P(x)/Q(x)Proper Rational function = P(x)/Q(x)Rational Functions = FRational Functions = F11(x) + F(x) + F22(x) + … + F(x) + … + Fnn(x)(x)

P(x)/Q(x)= FP(x)/Q(x)= F11(x) + F(x) + F22(x) + … + F(x) + … + Fnn(x)(x)FF11(x) + F(x) + F22(x) + … + F(x) + … + Fnn(x) are in the forms of (x) are in the forms of A/(ax+b)A/(ax+b)kk or (Ax+B)/(ax or (Ax+B)/(ax22 + bx + c) + bx + c)kk Denominators are factors of Q(x)Denominators are factors of Q(x)Sum is called the partial fraction decomposition of P(x)/Q(x) and the terms Sum is called the partial fraction decomposition of P(x)/Q(x) and the terms

are called partial fractions.are called partial fractions.Determine partial fraction decomposition: determining the exact form of thDetermine partial fraction decomposition: determining the exact form of th

e decomposition and finding the unknown constants.e decomposition and finding the unknown constants.

8.5 continue8.5 continue

• Linear Factor RuleLinear Factor RuleFor each factor of the form (ax+b)For each factor of the form (ax+b)mm the partial Fraction d the partial Fraction d

ecomposition contains the following sum of m partial ecomposition contains the following sum of m partial fractionsfractions

AA11/(ax+b) + A/(ax+b) + A22/(ax+b)/(ax+b)22 + … + A + … + Amm/(ax+b)/(ax+b)mm

AA1 1 AA2 2 AAm m are constants to be determined.are constants to be determined.

8.5 continue8.5 continue

Quadratic Factor RuleQuadratic Factor RuleFor each factor of the form (axFor each factor of the form (ax22+bx+c)+bx+c)mm the partial fractio the partial fractio

n decomposition contains the following sum of m partn decomposition contains the following sum of m partial fractionsial fractions

(A(A11x + Bx + B11)/(ax)/(ax22+bx+c) + (A+bx+c) + (A22x + Bx + B22)/(ax)/(ax22+bx+c)+bx+c)22 + … + (A + … + (Ammx + x + BBmm)/(ax)/(ax22+bx+c)+bx+c)mm

Examples~Examples~

∫∫(4x(4x22 + 13x – 9)/(x + 13x – 9)/(x33 + 2x + 2x22 - 3x) dx - 3x) dx

= = (4x(4x22 + 13x – 9)/(x + 13x – 9)/(x33 + 2x + 2x22 - 3x) - 3x)= (4x= (4x22 + 13x – 9)/((x)(x+3)(x-1) + 13x – 9)/((x)(x+3)(x-1)=A/x + B/(x+3) + C/(x-1)=A/x + B/(x+3) + C/(x-1)4x4x22 + 13x – 9 = A(x+3)(x-1) + B(x)(x-1) + C(x)(x+3) + 13x – 9 = A(x+3)(x-1) + B(x)(x-1) + C(x)(x+3)Let x= 0 Let x = 1 Let x = -3Let x= 0 Let x = 1 Let x = -3A=3 C=2 B= -1A=3 C=2 B= -1∫∫(4x(4x22 + 13x – 9)/(x + 13x – 9)/(x33 + 2x + 2x22 - 3x) dx - 3x) dx= = ∫3/x dx - ∫1/(x+3) dx +∫2/(x-1) dx∫3/x dx - ∫1/(x+3) dx +∫2/(x-1) dx=3ln|x| - ln|x+3| + 2ln|x-1| + C =3ln|x| - ln|x+3| + 2ln|x-1| + C

Section 8.7 Numerical Section 8.7 Numerical Integration; Simpson’s RuleIntegration; Simpson’s Rule

Riemann SumRiemann Sum

b nb n

∫∫f(x) dx = limf(x) dx = lim ∑ f(x ∑ f(xkk*)∆x*)∆x

a n->∞ k=1 a n->∞ k=1

Trapezoidal approximationTrapezoidal approximation

b b

∫∫f(x) dx = ((b-a)/2n)(yf(x) dx = ((b-a)/2n)(y00 + 2y + 2y11 + … + 2y + … + 2yn-1n-1 + y + ynn))a a

8.7 continue8.7 continue

Denoting errors of midpoint and trapezoidal approximationsDenoting errors of midpoint and trapezoidal approximations

bb

|E|Emm|=||=|∫f(x) dx - M∫f(x) dx - Mnn| Midpoint approximation error | Midpoint approximation error

aa

bb

|E|Ett|=||=|∫f(x) dx - T∫f(x) dx - Tnn| Trapezoidal approximation error| Trapezoidal approximation error

aa

8.7 continue8.7 continue

Theorem: Let f be continuous on [a,b], and let |ETheorem: Let f be continuous on [a,b], and let |Emm| and |E| and |Ett| be the | be the absolute errors that result from the midpoint and trapezoidal absolute errors that result from the midpoint and trapezoidal approximations of b using n subintervals approximations of b using n subintervals

∫∫f(x) dxf(x) dx aa1.) If graph is concave up or down on (a,b) 1.) If graph is concave up or down on (a,b) |E|Emm| < |E| < |Ett| | 2.) If graph of f is concave down on (a,b), then b2.) If graph of f is concave down on (a,b), then b

TTn n < < ∫f(x) dx < ∫f(x) dx < MMn n

aa

3.) If graph of f is concave up on (a,b), then b3.) If graph of f is concave up on (a,b), then b MMn n < < ∫f(x) dx < T∫f(x) dx < Tnn

a a

8.7 continue8.7 continue

Simpson’s rule – the combination of Trapezoidal and Midpoint Simpson’s rule – the combination of Trapezoidal and Midpoint approximations (best approximation for area)approximations (best approximation for area)

SS2n2n=(1/3)(2M=(1/3)(2Mnn + T + Tnn))

=(1/3)((b-a)/2n)[y=(1/3)((b-a)/2n)[yoo + 4y + 4y11 + 2y + 2y22 + 4y + 4y33 + 2y + 2y44 + … + y + … + y2n2n]]

8.7 continue8.7 continue

8.7 continue8.7 continue

Examples~Examples~

Use the Simpson Rule to find the areaUse the Simpson Rule to find the area22∫∫ee-x2-x2 dx =??? dx =???00

Section 8.8 Improper Section 8.8 Improper IntegralsIntegrals• Let Let ff be a function which is continuous on the closed be a function which is continuous on the closed

interval interval [a,  [a,  ∞∞)). We define . We define

If this limit exists and is finite then we say that the integral If this limit exists and is finite then we say that the integral

∞∞

∫ ∫ f(x) dxf(x) dxaa

is is convergentconvergent; otherwise, we say that the integral is ; otherwise, we say that the integral is divergentdivergent. .

8.8 8.8 ccontinueontinue

• Let Let ff be a function which is continuous on the closed be a function which is continuous on the closed interval interval ((∞∞, b], b]. We define . We define

If this limit exists and is finite then we say that the integral If this limit exists and is finite then we say that the integral

bb

∫∫f(x) dxf(x) dx- - ∞∞

is is convergentconvergent; otherwise, we say that the integral is ; otherwise, we say that the integral is divergentdivergent

8.8 continue8.8 continue

Let Let ff be a function which is continuous for all real numbers. If, for some real be a function which is continuous for all real numbers. If, for some real number number cc, both of , both of

Are convergent then we defineAre convergent then we define

and we say that the integral and we say that the integral

∞∞

∫ ∫ f(x) dx f(x) dx -∞-∞

is is convergentconvergent; otherwise, we say that the integral is ; otherwise, we say that the integral is divergentdivergent. .

Examples~Examples~

∞∞

∫∫dx/(xdx/(x22+9) =+9) =00 b b b b Lim ∫dx/(xLim ∫dx/(x22+9) = lim (1/3)tan+9) = lim (1/3)tan-1-1(x/3)|(x/3)|b-> b-> ∞∞ 0 b-> 0 b-> ∞∞ 0 0=.524=.524

00∫∫dx/(x-1)dx/(x-1)22 = =--∞∞

0 00 0Lim ∫x-1)Lim ∫x-1)-2-2 dx = lim -1/(x-1)| dx = lim -1/(x-1)|b-> -b-> -∞∞ b b-> - b b-> -∞∞ b b=1=1

QUIZ TIMEEEEEEEEQUIZ TIMEEEEEEEE

1. 1. ∫(x-1)/(x∫(x-1)/(x22-2x)-2x) dx = dx =a) (1/2)ln|x| + ln |x-2| + C c)ln|x-2| + ln|x| + Ca) (1/2)ln|x| + ln |x-2| + C c)ln|x-2| + ln|x| + Cb) (1/2)ln|(x-2)/x| + C d)(1/2)ln|(x-2)(x)| + Cb) (1/2)ln|(x-2)/x| + C d)(1/2)ln|(x-2)(x)| + Ce) None of thesee) None of theseAnswer : DAnswer : D2. 2. ∫(x∫(x33)(lnx) dx =)(lnx) dx =a) (xa) (x33)(3lnx + 1) + C c) (x)(3lnx + 1) + C c) (x44/4)(lnx – 1) + C/4)(lnx – 1) + Cb) (xb) (x44/16)(4lnx – 1) + C d) 3x/16)(4lnx – 1) + C d) 3x22(lnx – (1/2)) + C(lnx – (1/2)) + Ce) None of thesee) None of theseAnswer : BAnswer : B

3. 3. .785 .785 ∫ ∫coscos22x dx=x dx= 00a)1/2 c).643 a)1/2 c).643 b).393 d).893b).393 d).893e).143e).143Answer : CAnswer : C4. Using the midpoint area with (n = 3) and trapezoidal area (n=6) fin4. Using the midpoint area with (n = 3) and trapezoidal area (n=6) fin

d the area of the function y=6x-xd the area of the function y=6x-x2 2

a) Midpoint= 38 Trapezoid=35 c) Midpoint= 54 Trapezoid=60 a) Midpoint= 38 Trapezoid=35 c) Midpoint= 54 Trapezoid=60 b) Midpoint= 9 Trapezoid=30 d) Midpoint= 36 Trapezoid=36 b) Midpoint= 9 Trapezoid=30 d) Midpoint= 36 Trapezoid=36 e) Midpoint= 17.5 Trapezoid=17.5 e) Midpoint= 17.5 Trapezoid=17.5 Answer: AAnswer: A

5. Using Simpson’s rule, find the area for the function in the previo5. Using Simpson’s rule, find the area for the function in the previous questionus question

a) 38 c) 28a) 38 c) 28b) 37 d) 30b) 37 d) 30e) 112e) 112Answer : BAnswer : B6. 6. ∫ (e∫ (exx) (cosx) dx) (cosx) dxa) (1/2)ea) (1/2)exx(cosx + cosx) + C c) e(cosx + cosx) + C c) ex x sinxsinxb) (1/2)eb) (1/2)exx(sinx + cosx) + C d) e(sinx + cosx) + C d) ex x sinx + esinx + ex x cosxcosxe) (1/2)ee) (1/2)exx(cosx + sinx) + C(cosx + sinx) + CAnswer = BAnswer = B

7. 7. ∞∞ ∫ ∫dx/(1+xdx/(1+x22) =) = --∞∞a) 3.142 c) 0a) 3.142 c) 0b) 1.571 d) .785b) 1.571 d) .785e) None of thesee) None of theseAnswer : AAnswer : A8. ∫1/(x8. ∫1/(x22√16-x√16-x22) dx) dxa) (√16-xa) (√16-x2 2 )/16x c) 16x/(√16-x)/16x c) 16x/(√16-x2 2 ))b) 16x d) -(√16-xb) 16x d) -(√16-x2 2 )/16x )/16x e) None of thesee) None of theseAnswer : DAnswer : D

9. 9. ∞∞ ∫∫1/x1/x2 2 dx =dx = 11a) 0 c) 1.5a) 0 c) 1.5b) 2 d) -1b) 2 d) -1e) None of thesee) None of theseAnswer : E, real answer is 1Answer : E, real answer is 110. ∫ (√4-x10. ∫ (√4-x22)/x)/x2 2 dx =dx =a) x/( √4-xa) x/( √4-x22) c) ( √4-x) c) ( √4-x22)/x) )/x) b) sinb) sin-1-1(x/2) d) -( √4-x(x/2) d) -( √4-x22)/x) – sin)/x) – sin-1-1(x/2) + C(x/2) + C

e) ( √4-xe) ( √4-x22)/x) – sin)/x) – sin-1-1(x/2) + C(x/2) + C

Answer : EAnswer : E

BibliographyBibliography

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