1
INTEGRATION of FUNCTION of ONE VARIABLE
INDEFINITE INTEGRAL
Finding the indefinite integrals
Reduction to basic integrals, using the rule ....)()( =⋅′∫ dxxfxf n
1. ( )dxxxx∫ +3 2. ∫
+dx
x
x
3
2)32(
3. dxx∫ +100)32(
4. dxx
∫+ 32
1 5. dx
xx∫
++ 22
1
2 6. dx
xx
x
∫++
+
22
1
2
7. dxxx
x
∫++ 22
2 8. dx
xx∫
++ 22
1
2
9. dx
x
x
∫+1
2
10. dx
x∫
−
29
1 11. ∫
+
dx
xx2
2
1 12. dx
x
xxx
∫+
2
13. ∫+
dxx
x
122
14. dxxx902 )12( +∫ 15. dxxx
243 )1( −∫
16. dxx
x∫
+
2sin
12sin
2 17. xdx∫
2sin 18. xdx∫3
sin
19. ∫ xdxtan 20. xdx∫2
tan 21. xdxex
cosh∫
22. dxxx∫2sin 23. dx
x
x
∫ln
24. dxxx
∫ln
1
25. dxxx
3/2)(ln1∫ 26. dxee
xx 10)32( +∫ 27. dxe
e
x
x
∫+ 32
28. dxx
x
21
∫ 29. dxx
x
∫+
+
)12(cos
)12sin(2
30. dx
x
x
∫−
2
2
1
)(arcsin
31. ∫ dxx)4tanh( 32. dxx)3(sinh 2
∫ 33. dx
x∫
+2
61
1
34. dx
x
x
∫−
−
21
1 35. dx
xx
xx
∫−
−+
1
12 36. ∫
+
dxxcos1
1
Right or wrong?
1. cxx
xdxx +=∫ sin2
sin
2
2. ∫ +−= cxxxdxx cossin
3. ∫ ++−= cxxxxdxx sincossin 4. cx
dxx ++
=+∫ 3
)12()12(
3
2
5. cxdxx ++=+∫32 )12()12( 6. c
xdxx +
+=+∫ 6
)12()12(
3
2
Answers:
1. c
x
x ++
45
24
2/5 2. c
xx
x +−−2
2
912ln4 3. c
x+
+
202
)32( 101
4. c
x
+
+
2
32ln
2
5. cx ++ )1arctan( 6. c
xx
+++
2
)22ln( 2
7. cx
xx
++−++
)1arctan(2
)22ln( 2
8. cxarsh ++ )1( 9. cx ++12 10. c
x
+
3arcsin 11. cxarch ++ )1(
12. c
x
x +−2
2 13. c
x
++
4
)12ln( 2
14. c
x
++
364
)12( 912
15. c
x
+−
12
)1(34
16. c
xx
+−−
2
2cot
2
2cos 17. c
xx
+−
4
2sin
2 18. c
xx ++−
3
coscos
3
19. cx +− cosln
20. cxx +−tan 21. c
xex
++
24
2
22. c
x
+−
2
cos2
23. c
x
+
3
)(ln2 2/3
24. cx +lnln
25. c
x
+
5
)(ln3 3/5
26. c
ex
++
22
)32( 11
27. c
ex
++
2
)32ln( 28. c
x
+⋅ 22ln
2
29. c
x
+
+ )12cos(2
1 30. c
x
+
3
)(arcsin 3
31. cx +)4cosh(ln4
1 32. cx
x
+
−
6
)6sinh(
2
1
33. cx
+
−
6
)6(sinh 1
34. cxx +−+2
1arcsin 35. cxx ++− ln212 36. c
x
+
2tan
Right or wrong? 1. w 2. w 3. r 4. w 5. w 6. r
Integration by parts
∫ ∫ ′−=′ dxxgxfxgxfdxxgxf )()()()()()(
1. dxexx
∫−
+ )12( 2. ∫ − dxxx )13cos( 3. ∫ arctgxdx
4. ∫ + xdxx ln)1( 5. dxx )1ln( 2+∫ 6. xdxe
x
cos∫
Answers:
1. cexx
++−−)32( 2. c
x
x
x
+−
+−
9
)13cos()13sin(
3 3. c
xxarctgx +
+−
2
)1ln( 2
4. cx
x
xx
x
+−−
+
4ln
2
22
5. carctgxxxx ++−+ 22)1ln( 2
6. cxxe
x
++
2
)cos(sin
Integration of rational functions
1. ∫+−
dxxx
x
232
3
2. dxx
x
∫+12
4
3. dxxxx
xx
∫−+
++
)2)(2(
322
4. dxxx
x
∫−
+
)2(
322
5. dxxx
x
∫+
−
)4(
122
2
6. dxxxx
xx
∫++
++
)1)(1(
53222
2
7. dxx
x
∫+1
8. dxx
x
∫+12
2
9. dxx
x
∫−1
22
3
3
Answers:
1. cxxx
x
+−+−−+ 2ln81ln32
2
2. cxx
x
++− arctan3
3
3. cxxx +−+++− 2ln8
132ln
8
9ln
4
3 4. cx
x
x +−++− 2ln4
7
2
3ln
2
7
5. cxx +++− )4ln(8
9ln
4
1 2 6. cxx
x
x +−++−− arctan31ln25
ln2
7. cxx ++− 1ln 8. cxx +− arctan 9. cxx +−+ 1ln22
Integration by substitution
1. dxx∫ −
216 2. dxxx∫ − 2
2 3. dx
xx∫
+22
1
1
4. dxxx∫ + 3 5. dxx
∫++ 121
1 6. dx
xx∫
+
1
7. dxe
e
x
x
∫+
−
1
12
2
8. dxee
e
xx
x
∫++
+
34
1
2 9. dx
x
x
∫−
4
21
10. dxx
∫+ sin1
1 11. dx
x∫
+ sin2
1 12. dx
x
x
∫− cos1
cos
Answers:
1. c
xxx
+
−+
2
41
48
4arcsin8 2. c
xar
xx +−
−−−−
2
)1cosh(1)1()1(
2
1 2
3. c
x
x
++
−
21
4. cxx ++−+2/32/5
)3(2)3(5
2
5. cxx +++−+ )121ln(12 6. cx ++ )1ln(2
7. cxex
+−+ )1ln( 2 8. c
xex
+++
−
33
)3ln( 9. cx +− )(arcsincot
3
1 3
10. c
x
+
+
−
2tan1
2 11. c
x
++
3
1)2/tan(2arctan
3
2 12. cxx +−− )2/cot(
DEFINITE INTEGRALS
Express the limits as definite integrals:
1. ∑=
→
∆
n
k
kkP
xc
1
2
0
lim , where P is a partition of [ ]2,0 .
2. k
n
k kP
x
c
∆∑=
→
10
1lim , where P is a partition of [ ]4,1 .
3. k
n
k
kP
xc ∆−∑=
→
1
2
0
4lim , where P is a partition of [ ]1,0 .
4
Answers:
1. dxx∫2
0
2 2. dxx∫4
1
1 3. dxx∫ −
1
0
24
Find the derivative of
1. ∫x
tdt
0
cos 2. dtt
x
∫ +
0
21 3. dtt
x
∫2
0
cos
4. dt
t
x
∫−
sin
02
1
1 5. dt
t
x
∫+
2
0
61
1
Answers:
1. x
x
2
cos 2. 2
1 x+ 3. xx cos2 ⋅ 4. 1 5. 12
1
2
x
x
+
Find the average value of f over the given interval. At what point or points in the given
interval does the function assume its average value?
1. 1)( 2−= xxf , [ ]3,0 2. xxf sin)( = , [ ]π2,0 3. 1)( −= xxf , [ ] [ ]3,1,1,1−
Answers:
1. 0=avef , f(1)=0 2. 0=
avef , 0)( =πf 3. [ ] 2/1,1,1 −=−
avef ,
aveff =± )2/1( ,
[ ] 1,3,1 =avef , f(2)=1
Find upper and lower bounds for the value of
1. dxx
∫+
2/1
0
21
1 2. dxx )sin(
1
0
2
∫ 3. dxx∫ +
1
0
8
4. dxx∫ +
1
0
71 5. dx
x
ex
∫+
−100
0100
6. dxx∫ +
1
0
cos1
7. Suppose that f is continuous and that ∫ =
2
1
4)( dxxf . Show that f (x)=4 at least once on [ ]2,1 .
Answers: upper bound=ub, lower bound=lb
1. up=1/2, lb=2/5 2. ub=sin1, lb=0 3. ub=3, lb= 8 4. ub= 2 , lb=1
5. ub=1/100, lb= 200/100−
e 6. ub= 2 , lb=1 7. From the value of integral we get
4=avef . The function f is continuous, therefore f takes on 4 at least once on the given interval.
Evaluate the integrals:
1. dxxx )(
1
0
2+∫ 2. dx
x∫−
−
1
2
2
2 3. ∫ +
π
0
)cos1( dxx
4. dxxx
)11
(4
1
2/1
3−∫ 5. dx
x
x
∫−
4
9
1 6. dxxx 13
3
1
1
2+∫
−
5
7. dttt34
1
0
3 )1( +∫ 8. dttt3/1
7
0
2)1( +∫ 9. dx
x
x
∫− +
0
14
3
9
10. dxx
∫π
03
tan 11. dxx
x
∫−
+
2
1
21
4 12. dx
x
x
∫−
3/
0cos41
sin4π
13. dxe
e
x
x
∫+
4ln
01
2 14. dx
x∫
−
1
0
24
1 15. ∫
+
2
0
228 t
dt
16. dx
xx∫
−+
1
2/12
443
6 17. dx
xx∫
+−
4
2
2106
1 18. xdxe
x
sinh
2ln
0
∫
19. dxx
x
∫4
1
cosh8 20. dx
xx
e
e
∫2
ln
1
Answers:
1. 1 2. 1 3. π 4.-5/6 5. 3 6. 3/22/5 7. 15/16 8. 45/8 9. 2/)103( −
10. 3ln2 11. 2(ln5-ln2) 12. –ln3 13. 2(ln5-ln2) 14. π/6 15. π/16 16. π/2
17. π/2 18. (3-2ln2)/4 19. 16(sinh2-sinh1) 20. )12(2 −
APPLICATIONS of DEFINITE INTEGRALS
Area
1. Find the total area of the region between the curve and the X-axis
a) 23,22
≤≤−−−= xxxy b) 22,43
≤≤−−= xxxy
c) 81,3/1 ≤≤−= xxy
2. Find the area of the region enclosed by the curves
a) 2,22
=−= yxy b) xyxxy =−= ,22
c) xxyxy 4,22+−== d) 2,
2+== yxyx
e) 1,44 42=−=+ yxyx f) 3,2,4
22−=−−=−= xxxyxy
g) 3,2,42
=+−=−= xxyxy h) 2,,/1 === xxyxy
3. Find the area of the region bounded by the curves
a) y = 1/x, y = 5/2 – x b) xxy 22+= , 24 xy −=
c) 2yx = , 4/31 2yx += d) xyxxy 3),1( =−=
e) π/2,sin xyxy == f) 53,)1( 2−=−= xyxy
4. The region bounded below by 2xy = and above by y=4 is to be partitioned into two
subsections of equal area by cutting across it with the horizontal line y=c. Find the value of c.
5. Determine the area of the region enclosed by the Y-axis, the graph of xy = and its tangent
line touching the curve at the point whose abscissa is 4.
6. Find the slope of the line y=mx ( m positive number), if the area of the region enclosed by this
line and the graph of 2xy = is equal to 36.
Answers:
1.a) 28/3 b) 8 c) 51/4 2.a) 32/3 b) 9/2 c) 8/3 d) 9/2 e) 104/15 f) 11/3
g) 11/6 h) 3/2-ln2 3.a) 15/8-2ln2 b) 9 c) 8/3 d) 32/3 e) 1-π/4 f) 1/6
6
4. 23 )4(=c 5. 2/3 6. m=6
Volume
1. Rotate the given curve about the X-axis and determine the volume of the generated solid
a) xy /1= , [ ]3,1∈x b) 2
cos1x
y += , [ ]2/,2/ ππ−∈x c) xxey = , [ ]1,0∈x
2. Rotate the given curve about the Y-axis and determine the volume of the generated solid
a) 2
1
1
xy
+
= , [ ]1,2/1∈y b) xy ln= , [ ]2/3,2/1∈y c) xey = , [ ]2,1∈y
3. Rotate the graph of the function 2/3xy = ( axy ≤≤≥ 0,0 ) about both the Y-axis and the X-
axis. What is the value of a if both solids have the same volume?
Answers:
1. a)3
2π b) )22
4
3(2 +π
π c) )1(4
2−e
π
2.a) )2
12(ln −π b) )(
2
3 ee −π
c) )22ln42ln2( 2+−π 3. a=144/49
Arc length
1. Calculate the arc length of the curve
a) 21,2
1
6
3
≤≤+= xx
xy b) 40,
2/3≤≤= xxy
c) 22,4
1
82
4
≤≤+= yy
yx d) 90,
5
4 4/5≤≤= xxy
2. Find the value of b knowing that the arc length of the curve segment given by the graph of the
function 19
2)(
2/3+= xxf and lying between the points a=0 and b is equal to 42 units.
3. Find the arc length of
a) 2ln2ln,2
≤≤−+
=
−
xee
y
xx
b) dtty
x
∫=0
2cos , 4/0 π≤≤ x
Answers:
1.a) 17/12 b) )110(27
8 2/3− c) 25/16 d) 232/15 2. b=27 3.a) 3/2 b) 1
IMPROPER INTEGRALS
A) Evaluate the integrals:
1. dxx∫∞
1
001.1
1 2. ∫
∞−+
2
24
2
x
dx 3. dx
x∫−
1
1
3/2
1 4. ∫
∞
∞−+
22 )1(
2
x
xdx 5. dx
x
x
∫∞
+0
21
arctan 6. dxxe
x
∫∞−
0
7. ∫−
2
0
24 x
dx 8. ∫
−
2
0 1x
dx 9. dxxe
x
∫∞
∞−
−2
2 10. ∫1
0
ln xdx 11. ∫∞
−2
21
2
x
dx 12. ∫
2/
0
tan
π
xdx
13. ∫∞
++0
2 )1)(1( xx
dx 14. ∫
∞
−++
1
2 65xx
dx 15. dx
x
ex
∫−1
0
7
B) Test the integrals for convergence (integration, comparison test)
1. dxexx/1
2ln
0
2 −−
∫ 2. ∫+
π
0 sin xx
dx 3. ∫
∞
+1
31x
dx 4. ∫
∞
+01 x
e
dx 5. ∫
∞
+06
1x
dx
6. ∫∞
2ln x
dx 7. dx
x
ex
∫∞
1
8. dxx
x
∫∞
+
π
cos2 9. ∫
∞
∞−+14
x
dx
C) Find the values of p for which each integral converges
1. ∫2
1)(ln pxx
dx 2. ∫
∞
2)(ln pxx
dx
Answers:
A) 1. 1000 2. 3π/4 3. 6 4. 0 5. 8/2
π 6. 0 7. π/2 8. 4 9. 0 10. -1
11. ln3 12. +∞ 13. π/4 14. ln2 15. 2(1-1/e)
B) 1. convergent, the value of the integral is 2ln/1−e 2.
xxx
1
sin
10 ≤
+⟨ , dx
x∫π
0
1 is
convergent therefore the original integral is convergent 3. 33
1
1
10
xx
⟨+
⟨ , dxx∫+∞
1
3
1 is
convergent therefore the original integral is convergent 4. x
x
e
e
−
⟨+
⟨1
10 , dxe
x
∫+∞
−
0
is
convergent therefore the original integral is convergent 5. convergent 6. divergent ,
01
ln
1⟩⟩
xx
, dxx∫+∞
2
1 is divergent 7. divergent 8. divergent 9. convergent
C) 1. convergent if p<1 , divergent if p≥1 2. convergent if p˃1 , divergent if p≤1