Created by T. Madas Created by T. Madas DEFINITE INTEGRATION MIX
Created by T. Madas
Created by T. Madas
DEFINITE
INTEGRATION
MIX
Created by T. Madas
Created by T. Madas
Part 1
Created by T. Madas
Created by T. Madas
1. 2
0
11
4 1dx
x=
+∫
2. 2
0
sin 2 1x dx
π
=∫
3. 6
0
3sin 4
6 4x dx
π
π + =
∫
4. 2 2
0
sin4
x dx
π
π=∫
Created by T. Madas
Created by T. Madas
5. 2
3
1
15ln 4ln 2
16x x dx = −∫
6. ( )
12
20
1 3ln
3 42
xdx
x
= +−∫
7. ( )
2
2
1
2 ln 27
122 1
xdx
x
+=
−∫
Created by T. Madas
Created by T. Madas
8. ( )( )
1
0
3ln 2
1 2
xdx
x x= −
+ −∫
9. ( )4 2
0
1tan 4
4x dx
π
π= −∫
10. 2
0
2 17
64 1
xdx
x
+=
+∫
Created by T. Madas
Created by T. Madas
11. ( )1
2 2
0
1e 1 3e4
xx dx
− −= −∫
12. ( )2
20
24 2 1
4
xdx
x
= −
+∫
13. ( )( )
13
16
14 1 53ln
2 1 1 4
xdx
x x
+ =
+ − ∫
Created by T. Madas
Created by T. Madas
14. ( )
36
0
1ln16
2dx
x x
=+∫
15. ( )4
0
312 cos 2 22
x x dx
π
π= −∫
16. ( ) ( )2 2
0
12sin 3cos 13 244
x x dx
π
π− = −∫
Created by T. Madas
Created by T. Madas
17. 2
4
4 sin 2 1x x dx
π
ππ= −∫
18.
32
6
9
24 2
xdx
x−
= −−∫
19. ( )
32
5
1
12
2 1
xdx
x
+=
−∫
20. ( )
112
2
0
16sin 34
d
π
θ θ π= −∫
Created by T. Madas
Created by T. Madas
21. ( )( )
12
2
0
1 1 1ln 3
6 41 1dx
x x
= +− +∫
22. ( )
14
2
0
1sec ln 44
x x dx
π
π= −∫
23. ( )
1
4
0
9 13
92 1dx
x
=+∫
Created by T. Madas
Created by T. Madas
24. ( )12
2 2
0
18 sin 42
x x dx
π
π= +∫
25.
14
0
12 cos44
x x dx
π
= −∫
26. ( ) ( )
14
2
0
5cos sec 28
x x dx
π
π+ = +∫
Created by T. Madas
Created by T. Madas
27.
ln 5 2
ln 2
3e20
e 1
x
x
dx =
−∫
28. 3 2
0
76
151
xdx
x=
+∫
29. ( )( )
6
2
5 3ln54
2 3 2
xdx
x x
+=
− +∫
Created by T. Madas
Created by T. Madas
30. ln 2
0
4 e 2 ln 4x
x dx−
= −∫
31. 3
20
1ln 2
29
xdx
x
=+∫
32.
14
0
2sin 2
4 2x dx
ππ
+ = ∫
33. 23
1
1 1 ln 36 42 1
xdx
x= +
−∫
Created by T. Madas
Created by T. Madas
34. ( )( )
4
0
13 24ln 3 3ln 2
4 2 1
xdx
x x
−= −
+ +∫
35. e
1
ln 1x dx =∫
36.
13
16
1cos33
x dx
π
π
= −∫
37. ( )
12
3
0
4cos 1 sin 15x x dx
π
+ =∫
Created by T. Madas
Created by T. Madas
38.
16
18
2 1 1 1cot 2 32 6 24
x dx
π
π= − −∫
39. ( )( )
12
0
3 5 4 ln 231 2 3
xdx
x x
−=
− −∫
40. ( ) ( )ln 4
22
ln 2
e 2 4 9 ln 2x
dx− = +∫
41.
12
0
1sin 24
x x dx
π
π=∫
Created by T. Madas
Created by T. Madas
42. 1 2
21
9 42 3ln 5
9 4
xdx
x−
+= − +
−∫
43. 7 2
1
652
152
xdx
x−
=+∫
44. 2
0
6ln16
3 2dx
x=
+∫
45. ( )5
2
1 52 8ln64 1
dxx
= ++ −∫
Created by T. Madas
Created by T. Madas
46. ( )
14
20
cos2 1 22cos
xdx
x
π
π= −∫
47. 4
0
2cos 3
4 6x dx
π
π + = −
∫
48. ( )
4
3
2
8 5
163 4dx
x
=−∫
49.
52
1
4 20
32 1
xdx
x=
−∫
Created by T. Madas
Created by T. Madas
50. 3
0
3cos 3
3 3x dx
π
π + = −
∫
51. ( )( )
12
2
0
18 4 7 3ln 23 2
4 3 1
x xdx
x x
− −= +
− +∫
52. ( ) ( )2 2
4
1sin cot 26 4 28
x x dx
π
ππ+ = − −∫
53. 3 3
0
3tan ln 22
x dx
π
= −∫
Created by T. Madas
Created by T. Madas
54. 1
21e
1 3ln 1
4 ex x dx
= −
∫
55. ( )
1
2
0
1ln 2
21
xdx
x
= −+∫
56. ( )( )
32
2
2
4 91 ln 2
4 1
x xdx
x x
− += +
− −∫
Created by T. Madas
Created by T. Madas
57. ( )12
0
110sin8 cos 2 16 3 3
12d
π
θ θ θ = +∫
58. 3
6
3sin 4
6 8x dx
π
π
π + = −
∫
59. ( ) ( )e
2 3
1
21 ln e 59
x x dx+ = +∫
Created by T. Madas
Created by T. Madas
60. ( )
5
3 2
3
1 2cos 4 3 3x dx
π
ππ− = +∫
61. 3
0
1161
15x x dx+ =∫
62. 3 2
22
2 251 ln
182 3
x xdx
x x
+ + = +
+ − ∫
63. ( )
1
2
0
93
2 1dx
x
=+∫
Created by T. Madas
Created by T. Madas
64. 6
0
3sin sin 3
16x x dx
π
=∫
65. ( )2
3 2
0
1ln 2 ln 22
x x dx+ = +∫
66. ( )( )
14
0
4ln 3
1 2 1 2dx
x x=
+ −∫
Created by T. Madas
Created by T. Madas
67. 2 3
0
2cos
3x dx
π
=∫
68. 3
0
4ln9
2 3dx
x=
+∫
69. ( )2
3
20
64 1 5
1
xdx
x
= +
+∫
70. 8 2
25
26 4ln 3
16
xdx
x
= +−∫
Created by T. Madas
Created by T. Madas
71. ( )2
2
1
ln 1ln 2
2
xdx
x=∫
72. ( )( )
1
2
0
17 5 1 10ln
2 33 2 2
xdx
x x
− = +
+ −∫
73. ( ) ( )0
1cos 2 2 4 164
x x dx
π
π= + −∫
74. ( )12
4
2
0
e 2 e 1x
dx = −∫
Created by T. Madas
Created by T. Madas
75. ( )
92
2
4
5 8 1 32 5ln
3 242 1
x xdx
x x
− + = −
−∫
76. ( )( )
1
0
3ln 4
2 1dx
x x
= −− +∫
77. 3
0
11 3
1 sindx
x
π
= +−∫
Created by T. Madas
Created by T. Madas
78. ( )( )
62
2
2 114 4ln 3 3ln 2
2 2 3
x xdx
x x
− += + −
+ −∫
79. 0 2
1
1ln 2
1 2
xdx
x−
= − +−∫
80. 100
0
140ln 2 20
20dx
x= −
−∫
Created by T. Madas
Created by T. Madas
81. 1 2
20
1 ln 34
xdx
x
= −−∫
82. ( )6 3
0
32 1sin 3 16 9 33 8 24
d
π
θ θ = − = −∫
83. ( )ln 2
0
1 4ln31 e
xdx =
+∫
Created by T. Madas
Created by T. Madas
84. 4
2 1 3
0
e 2ex
dx+
=∫
85. 1 3
0
5 ln 261
xdx
x= −
+∫
86. ( )( )( )
1
0
103ln 3 3ln 2
1 3 2 1dx
x x x= −
+ + +∫
Created by T. Madas
Created by T. Madas
87. ( )22
0
11 tan 2 ln 42
x dx
π
+ = + ∫
88. ( )3
0
sin 2 31 2ln41 cos
xdx
x
π
= ++∫
89. ( )2
0
153 ln 1 ln 46
x x dx+ = − +∫
Created by T. Madas
Created by T. Madas
90. 0
14 1
2 1 430
x x dx− =∫
91. ( ) ( )e
2
1
11 ln e 34
x x dx− = −∫
92. 0
213
1 1ln 3
123 6 9dx
x x−
=− −∫
Created by T. Madas
Created by T. Madas
93. ( )2 2 2
0
1sin 4
16x x dx
π
π= +∫
94. ( ) ( )e
2 2
1
1ln e 14
x x dx = −∫
95. ( )2 5
0
107sin cos 1 sin
14x x x dx
π
+ =∫
96. 5 2
2
356
151
xdx
x=
−∫
Created by T. Madas
Created by T. Madas
97. ( )( )( )
( )5
0
1 8ln71 2 3
dxx x x
=+ + +∫
98. ( )( ) 2
0
1 3 sin 2 10x x x dx
π
π π− + = + −∫
99. ( ) ( )0
1
33ln 2 3 ln 27 22
x dx
−
+ = −∫
Created by T. Madas
Created by T. Madas
100. 6 3
0
12sec 4 3ln 3x dx
π
= +∫
101. ( )2 3 2
5
4 51 ln3
xdx
x
+= +∫
Created by T. Madas
Created by T. Madas
102. ( ) ( )1
e2 2 2
e
1ln 1 e 3e
4x x dx
−
− − = − + ∫
103. 1 2
20
141
xdx
x
π= −
+∫
104. 2 cos
0
e sin cos 1x
x x dx
π
=∫
Created by T. Madas
Created by T. Madas
Part 2
Created by T. Madas
Created by T. Madas
1.
2 2
20
124
xdx
x
π= −
−∫ , use 2sinx θ=
2. ( )2
2 21
1 13 1
44
dx
x x
= −
−∫ , use 2cosx θ=
3.
( )( )
1
22
0
1 12
81
dx
x
π= +
+∫ , use tanx θ=
Created by T. Madas
Created by T. Madas
4. ( )2
2 22
1 13 2
21
dx
x x
= −
−∫ , use secx θ=
5.
34
20
1
63 4
dx
x
π=
−∫ , use 3
sin2
x θ=
6.
( )32
1
2
0
1 1
21 3
dx
x
=
+∫ , use 1
tan3
x θ=
Created by T. Madas
Created by T. Madas
7.
1
20
1
42
dx
x
π=
−∫ , use 2 sinx θ=
8.
12
20
1 3
364 3dx
x
π=
+∫ , use 3
tan2
x θ=
9.
( )32
1
2
0
1 3
124
dx
x
=
−∫ , use 2sinx θ=
Created by T. Madas
Created by T. Madas
10.
22
2
13 1
12
xdx
x
π−= − −∫ , use cosecx θ=
11.
1
20
1 3
94 3
dx
x
π=
−∫ , use 2
sin3
x θ=
12. 3 2
21
3 1121
xdx
x
π= − −
+∫ , use tanx θ=
Created by T. Madas
Created by T. Madas
13. ( )2
2
0
116 4 6 3
3x dx π− = +∫ , use 4sinx θ=
14.
( )
2
2
0
32
1 1
83 4
dx
x
=
+∫ , use 2
tan3
x θ=
15. 2
2
0
8 316 3 2
9x dx
π− = +∫ , use
4sin
3x θ=
Created by T. Madas
Created by T. Madas
16.
( )
3
22
0
27 1
8 49
dx
x
π= +
+∫ , use 3tanx θ=
Created by T. Madas
Created by T. Madas
Part 3
Created by T. Madas
Created by T. Madas
1. ( )8
2
4
16 16 3 8ln 2 3x dx− = − +∫
2. ( )
1
0
1
21dx
x x
π=
+∫
3. 3
6
2sec ln 3 1
3x dx
π
π
= +∫
4. ( )1
3 2
0
21 2 1
15x x dx+ = +∫
5. ( )
ln3
ln 2
cosh 1 5
sinh cosh 1 2
xdx
x x
+=
−∫
6. ( )3
1
5 1arctan 1 3
12 2x x dx
π= + −∫
7. ( )2
53
43
1 11 ln 2
39 16
xdx
x
+= +
−∫
8. 2
52
32
94 9 5 ln3
4x dx− = −∫
9. ( )4
0
9arsinh ln 2 5 5
2x dx = + −∫
10. ( )( )1 2
20
12 ln 1 2
21
xdx
x
= − ++∫
11. 2
7
5
1
10 29 8dx
x x
π=
− +∫
12. ( )2
7
5
1ln 1 2
10 29dx
x x
= +− +∫
Created by T. Madas
Created by T. Madas
13. 2
7.5
2.5
1803
4 75dx
xπ=
+∫
14. 2
3
2
1
33 2dx
x x
π=
+ −∫
15. 2
7
5
1 1ln 2
9 2 12
xdx
x
π+= +
+∫
16. 2 2
3
0
1
9cos sin 18dx
x x
ππ
=+∫
17.
12
ln3
0
sech6
x dxπ
=∫
18.
1
3
4
0
4 3 1ln
1 33 1dx
x
π += + − − ∫
19.
3
2
2
0
8
4 9 3dx
x
π=
+∫
20. ( )( )
1
2
0
10 16 1ln arctan
5 21 4dx
x x
= +
+ + ∫
21. ( )( )
3
2
0
8 26 1ln 2 arctan arctan1
25 22 4
xdx
x x
= + −
+ + ∫
22. ( )
9
4
0
1
39dx
x x
π=
−∫
23. ( )( )
( )( )
0
2
1
1 22ln 2
41 1
x xdx
x x
π
−
+ += −
− +∫
24. ( )2
0
26
1ln 2 1
1 3cos3dx
x
π
= −+∫
Created by T. Madas
Created by T. Madas
25. ( )1
21
1ln 1 2
2 5dx
x x−
= ++ +∫
26. 3
4
09 24
xdx
x
π=
+∫
27.
5
6
5
3
2
1 2
925 9dx
x
π
−
=−∫
28. ( )2
2
1
1 12 2 ln 1 2 2
2 2x x dx− + = + −∫
29. 2
31
2 1arccos
3 39
xdx
x
=−∫
30. 4 2
20
sec
43 sec
xdx
x
π
π=
−∫
31.
( )3
2
2
20
1 12
84
dx
x
=
+∫
32. ( )2 2
21
12 ln 2 1
21
xdx
x
= − + +∫
33. 2
2
0
36 32 3 ln 2
4 3
xdx
xπ
−= −
+∫
Created by T. Madas
Created by T. Madas
34. [ ]
1
2
0
12
1 4
xdx
xπ= −
−∫
35. ( )
4
1
1 2 3arctan
3 119dx
x x
=
+ ∫
36. 2
0
1 3
1 sin 9dx
x
π
π=+∫
Created by T. Madas
Created by T. Madas
37. 12
ln3
ln 3
1
5cosh 4sinh 18dx
x x
π=
−∫
38. ( ) ( )12
0
1arsinh 2 3 ln 2 3 22
x dx = + −∫
Created by T. Madas
Created by T. Madas
recognition/substitution
partial fractions
parts
trig identities
mixed tech
antiderivatives/linear adjustment