Indefinite Integration

Exercises
Level - 1
(Problems Based on Fundamentals)

ABC of Integration

1. Evaluate :  log x xdx

 3 
log x 2
2. Evaluate :  2log x 3

 m x m m
3. Evaluate :   x  m  m   dx
 x

4. Evauate :  2x.3x.dx

5. Evaluate :  tan 2 xdx

6. Evaluate :  cot 2 xdx

dx
7. Evaluate :  sin 2 x cos2 x

1
 Indefinite Integration

8: Evaluate: 29. Eva 1uate..

• 9. Evaluate: f (tan +cotx) dx

2
x
30. Evaluate: f (cost-cos2x
- cosx )dx
10. Evaluate:
31. Evaluate: f (tanx +cotx dx+secx +cosx)
11. Evaluate: f ( 310g 5; - i0g )dx 5x

Type 2
1 2. Evaluate: f (1+ tan(;-x))(1+tan(; +x))dx 32. Evaluate: f �l dx

13. Evaluate: f
(8 1+x
+4 l+x dx
. ) 33. Evaluate:
x+
f (1 + x)2 dx
22x x(l + X)
2

14. Evaluate: f (� +; +� +mx )dx

34. Evaluate: f -2-- dx
x2 -2
X + 1
15. Evaluate:
f (ax +b x) 2 dx 1
ab
x x 35. Evaluate: f 213 x-113
(x +X +1) dx
(2x +3 x)2
16. Evaluate: f 2x ·3xdx 36. Evaluate:

17. If/'(x) = ¼ +.�

'11 -x
and/(1) = 1r , fi df( ).
2 2
n x
37. Evaluate: f ( x:-3 )dx
X +I
18. If J'(x) = acosx +b sinx and

find f(x). 2
(x6
38. Evaluate: ·f ---- dx
X + 1
I)
Type 1 39. Evaluate: f (x: ++ X: ++ 1I)dx
dx
X X
f 1-smx
f (-/- )dx
19. Evaluate: _
40. Evaluate:
20. Evaluate: J(si � x2 +co/x)dx +I
4
X

sm xcos x
f ( x: +x +1 dx
2
41. Evaluate: )
21. Evaluate: sin6x +cos6x dx +X +I
J( 2 )
X

f (x6-2--
+ I)dx
sin xcos 2x
42. Evaluate:
22 · Evaluate: X +I
Type 3
23. Evaluate: f ( cos4x-sin4x dx
-vl +cos4x
) 43. Evaluate: f sin- 1 (sinx)dx

f ( 1 +tan2x ) dx 4 4. Evaluate: f sin-1(cosx)dx

2
24. Evaluate:
(\11/1 +- cos2x
1 + cot x
45. Evaluate: ftan-1 cos 2 )c1x
25. Evaluate: f (cos(
-cos2x dx x

cosx ) -
46. Evaluate: f tan- 1 ( s in 2x )dx
f ( �x + : 4 + 2 dx
4
26. Evaluate: : ) 1 +COS 2X

27. Evaluate: f ( 5cos3� +.3�in3x )dx

cos x sm x

f ( cos
sin
28. Evaluate: x-sinx .
.) (l + sm2x)dx 48. Evaluate: f tan- 1 ( x )dx
.
cosx + smx I -cosx
2
 Indefinite Integration

. 49. Evaluate: f tan-1(.ff=smx)dx

YT+si�
71. Evaluate: J -V2x + 2014 � -V2x + 3013
I _ sinx )dx
50. Evaluate: tan 1 (
•
Type 6
1 + cosx
J -Vxx_ 1 dx
51. Evaluate: I tan-I (
72. Evaluate:
COSX

73. Evaluate: J
dx
x! 1
1 _ sinx )

52. Evaluate: I tan_ 1 - sinx)dx

dx

74. Evaluate: I
1(
cosx dx X
-V3x + 1
53. Evaluate: J tan-I (�(l + sinx) + �(1 - sinx)
�(l + sinx) _ �(1 + sinx)
) dx
75. Evaluate:

54. Evaluate: J tan-I (secx + tanx)dx x-l

55. Evaluate: I tan_

76. Evaluate: ...J dx
sin2x
1(
x+ 4

J x1-: 1 dx
)dx
1 + cos2x
77. Evaluate:
Type 4
J Type 7
56. Evaluate: (3x + 2)dx
I
78. Evaluate: cosx - sinx dx
57. Evaluate: J sinx + cosx

79. Evaluate: I 3cosx dx

lxdx_ 3

58. Evaluate: J � 2sinx + 5

80. Evaluate: I cosx - sinx dx

5 2x

59. Evaluate: J dx
ax+b
2 + sin2x

81. Evaluate: I xe + <!

e

60. Evaluate: J 3
x
dx4x+s .x dx 2
(
COS Xe )
61, Evaluate: J cos(5x + 3)dx
82 · Evaluate: Jx(l +�.nx. )2 .
62. Evaluate: J sin2xdx

63. Evaluate: J -V3x + 2 dx I

83. Evaluate: cosx - sinx + 1 - x dx
� + sinx + x
84. Evaluate: J
64. Evaluate: I dx 1 ! t!
-V3x + 4
85. Evaluate: J dx
Type 5 x(x3 + l)

65. Evaluate: J � 86. Evaluate: J ':

"1x+2 --lx+T
x(x + l)
66. Evaluate: J (f"ii+s� -v2x+3)
67. Evaluate: J
87. Evaluate:
Jx(x5� l)
� "'3x+l.
88. Evaluate. J. sin2x . dx
-ffx+4 . ·

68. Evaluate: J
sin5xsin3x
�
-v2x+3 ...f2x=3
69. Evaluate: J � J sin(x _ afsin(x - b)
89. Evaluate:
+ -{x

70. Evaluate: J 90 · Evaluate: J e -e dx

X -X
:
.../x+a ..../x+b e + e-x
x