DEFINITE INTEGRATION MIX - MadAsMaths · 2017. 3. 29. · DEFINITE INTEGRATION MIX . Created by T....
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Created by T. Madas Created by T. Madas DEFINITE INTEGRATION MIX
Transcript of DEFINITE INTEGRATION MIX - MadAsMaths · 2017. 3. 29. · DEFINITE INTEGRATION MIX . Created by T....
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
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