14 - Indefinite Integration
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PROBLEMS
14 - INDEFINITE INTEGRATION
Page 1
( Answers at the end of all problems ) Integrate the following functions with respect to x:
(1)
2 4 x + x + 1 2 x + 1
(4)
2 log 1 2
(7)
(2)
1
-
− x2 2
1 2
(8)
2
cos x sin x
1
( 10 )
( 11 )
x 1 - x3
( 13 )
( 16 )
( 19 )
( 22 )
( 25 )
1 + cos 2x (π /2 < x < π)
1
(5)
( 14 )
- sin 2x
( - π / 4 < x < π/ 4 )
( 17 )
cos 3 x
( 20 )
sin 2 x
x 1
( 23 )
- x3
1- x 1+ x
( 26 )
3
(x
(3)
x2 + 1
sin x 1 - sin x
(6)
1 2
2
1
- 1) 2 ( x - 2 ) 2
2
2
a cos x + b sin x
x2
- 1
(9)
( 12 )
x4 + 1
x tan
- 1 x2
x4 x2 + 1
e 2x
- 1
2x
+ 1
e
1 1 + tan x
sin x cos x 3 sin 2 x
1 + sin x (0 < x < π )
( 15 )
sin x sin 2x
( 18 ) cot x
1 sin ( x - a ) sin ( x - b )
1 5 + 4 cos x + 3 sin x
1 - cos x (0 < x < π )
6
( 21 )
sin 2 x cos 4 x
1
( 24 )
1
1 2 x
( 27 )
( 30 )
- x3 - x
5
x
2
( 29 ) sin x cos 2x
- 4 cos 2 x
sin x
1 + x4
cos x
1
( 28 )
x6 + 2
3x + 1 2x
2
+ x + 1
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 2
( Answers at the end of all problems )
1
( 31 ) x
x
- a
4
( 32 )
4
1 (x
- 1) ( 4 - x )
1
( 34 ) x 2ax - x 2 ( 0 < x < 2a ) 2
( 37 ) x e
( 40 )
( 43 )
3x
x
4
x
sin - 1 1
+
( 44 )
(1 +
- 1) ( x + 2 )
x2
x +
x - 1
sin 5x sin x
( 47 )
( 50 )
2
( 52 ) x sec x tan x
3 2
2
x
( 42 )
( 45 )
( 48 )
( x - a )2 ( x > 0, a > 0 )
x 2 tan - 1x
( 53 ) e - x sin 2 2x
8 - 2x - x 2
x
2ax
cos - 1
x
2
( 58 ) e
2x
sin 3x sin x
( x + 1)
2
ex
2
( 51 ) x sin x
1
( 54 )
4
- 1
( 57 )
x2 ( x3
- 1) ( x3 + 4 )
3
( 59 ) sec x
1 - x2
x2 + 1
4
( 56 ) sec x
- x2
1 + x2 (x<0)
x
( 55 ) 2 cos x
2x - 1
( 39 ) 5 e
x2
9 x2 )2
3x + 1
( 36 )
1
x3
( 46 )
( 49 )
( 3x
( 41 ) x sinx cos 3x
+ 1
x2
2
( 35 ) cos x sin 2x
( 38 )
x2
25 - x 2
( 33 )
( 60 )
x ( 3x
2
+ 2)(x - 2)
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 3
( Answers at the end of all problems )
( 61 )
( 64 )
( 67 )
x3 ( x2 + 2 ) ( x2 + 5 )
cos -1 x
- sin - 1 x x + sin -1 x
cos -1
x ( x 2 + a2 ) ( x 2 + b2 )
-
1 - sin x e ( 70 ) 1 + cos x (0 < x < π / 2 )
( 73 )
( 76 )
( 85 )
2
1 cos x ( 1 + 2 sin x )
x a - x
tan x 3 + 4 tan 2 x
1 + x2 1
( 88 )
( 65 )
( 68 )
1 6e
2x
+ 5e
x
+ 1
sin x cos x
1
( 63 )
( 66 ) x
x 2 + 4x - 1 x3 - x
- 1) ( x2 + 4 )
(x
( 69 )
2
cos ( log x )
x2 ( x 2 + 1) ( x 2 + 2)
x
( 79 ) sin x cos x ( sin 2x + cos 2x )
( 82 )
( 62 )
- x2 6x + 7
(x - 5)(x - 4)
1
( 71 ) x(x
( 74 )
( 77 )
( 80 )
3
( 72 )
+ 1)
1 sin x + sin 2x
x4 + x2 + 1 x2 + x + 1
sin x sin 4 x
( 75 )
tan x
sin x + cos x
( 81 )
sin 2x
1
( 83 ) ( ex +
3 2x 2 e )
1
( 86 )
(1 + x ) 1
4
- x2
( 84 )
( 87 )
2
( 89 ) sin 3x cos 3x
1- x
( 78 ) ( 1 + x )
( 90 )
x
a2 - x2 a2 + x2
1 ( 2x + 1 ) 4x + 3
1 cos 2x + 3 sin 2 x
sec x + 1 0 < x < π, x ≠ π / 2
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 4
( Answers at the end of all problems )
1
( 91 )
a - b 2 cos 2 x (a > b)
1
( 94 )
( 1 + x ) 1 + 2x
( 97 )
( 100 )
( 103 )
( 106 )
- x
( 115 )
( 118 )
cot x
2
2
(x + 4)
2
x + 1
1 sin 4 x + cos 4 x
1 sec x + cosec x
1 x ( xn + 1)
1+
x
1+
1 4 x
x a - x
tan- 1
tan x + tan 3 x 1 + tan 3 x
1 2
2 2
x (1 + x )
( 93 )
tan x 1 + tan x + tan 2 x
x
( 95 )
( 109 ) tanx tan2x tan 3x
( 112 )
tan x +
( 92 )
2
( 98 )
( 101 )
( 96 )
( 99 )
cot x
4e x + 6e - x 9e x - 4e - x
( 102 )
( 104 )
1 sin x + secx
( 105 )
( 107 )
x 1 + sinx
( 108 )
( 110 )
( 113 )
( 116 )
( 119 )
1 x3
x4 - 1
x ( x + 1) ( x + 2 ) ( x + 3)( x + 4)(x + 5)
x2 ( x sinx + cosx ) 2
sin - 1
x a + x
x2 x4
1
+
7x 2 + 1
+
sin x + cosx 9 + 16 sin2x
a 1-
x ax
tan-1 x
sin x
3
( 111 ) x sin x
( 114 )
( 117 )
( 120 )
a + x x xn -
2
1 - xn
1 5 - 2 cosx - 2 3 sinx
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 5
( Answers at the end of all problems )
sin ( x - a ) sin ( x + a )
( 121 )
( 122 )
1
( 124 )
sin 3 x sin ( x +
α)
x - α β - x
( 126 )
( 128 ) log log x +
4
1 + tan x
( 130 ) 1 ( sinx - 2cosx ) ( 2sinx + cosx )
( 131 )
( 133 ) x ( tan – 1x ) 2
( 134 )
sin x
( 136 )
( 123 )
( 125 )
1
( 127 )
3 + 2 tanx 4 + 3 tanx
1
cot -1 ( 1 - x + x 2 )
x
1+
x
sin 2 x 4 + cos 2 x
x7
( 129 )
( log x ) 2
-
1
x 12 - 1
( 132 ) sin 3 x ( 1 + cos 2 x ) 1 + cos 2 x + cos 4 x
1 4
(x
- 1 )3 ( x + 2 )5
- cosx 2
2
( sinx + cosx ) sinx cosx + sin x cos x
( 137 )
( 135 )
x3 (x
- 1) ( x - 2 ) ( x - 3 )
cos 2x sin x
Answers ( Add constant C to all answers )
Answer:
[1]
Answer:
x3 -1 + tan x 3
Answer:
[4]
sin - 1
x log 2
[2]
x5 5
Answer:
-
x3 + x + tan - 1 x 3
[3]
x3 - x + tan- 1 x 3
Answer:
Answer:
[ 5 ] sec x + tan x - x
[ 6 ] log ( e x + e – x )
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 6
( Answers at the end of all problems ) Answer:
Answer:
[ 7 ] tan x - cot x
[8]
Answer:
Answer:
1 - x3 - 1
1 log [ 10 ] 3
[ 13 ]
1 b tan- 1 tan x ab a
[ 11 ]
1 - x3 + 1
Answer:
Answer:
Answer: 1
log
2 2
x2 x
Answer:
- 2 sin x
1
[ 14 ]
Answer:
4
x 1 + log I cos x + sin x I 2 2
[9]
2
- 2x + 1 +
1
[ 12 ]
14
2x + 1
log I 3 sin 2 x
- 4 cos 2 x I
Answer:
( tan - 1 x 2 ) 2
[ 15 ] - 2 2
Answer:
sin (
π 4
x
-
2
1
) -
2
log tan (
π
-
8
x 4
)
Answer: 5
4 ( sin x ) 2 5
[ 18 ] -
[ 16 ] sin x + cos x
[ 17 ]
Answer:
Answer:
Answer:
[ 20 ]
[ 21 ]
1
[ 19 ] -
sin x
- sin x
Answer:
Answer:
sin -
2
[ 22 ]
1 sin ( x - a ) log sin ( a - b ) sin ( x - b )
3
3 1 2 x
1 192
( 12x
[ 23 ] tan
2 x 2
[ 24 ]
1 2 x2
1
[25]
[26]
[27]
Answer:
x - 2 x - 1
[29]
Answer:
2
x sec - 1 a
2
sin 2x x sin 4x 4 4 16
2
[32] 2 sin - 1
x
2
2
5 - x - 5 log
5 - x +
5
5 - x -
5
Answer:
Answer:
1 2a
4
+ 2 2 cos
Answer:
[28] 2
[31]
x
1
+ 3 Answer:
2 log tan
1
- 3 x 3 + 6 x 6 - 6 log ( 1 + x 6 )
Answer:
1 - x2
- 3 sin 4x + 3 sin 2x - sin 6x )
Answer:
Answer:
sin -1 x +
cot 5 x cot 3 x + - cot x - x 5 3
[30]
3 1 4x + 1 log 2x 2 + x + 1 + tan- 1 4 2 7 7 Answer:
x - 1 3
[33] -
25
- x2 x
- sin - 1
x 5
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 7
( Answers at the end of all problems ) Answer:
[34] -
Answer:
1
- x
2a
a
4
[35]
3
a
Answer:
Answer:
x2 2x 2 3x [37] + e 9 27 3
[38] 6x + 5 12
Answer:
Answer:
[39]
5 x e 2x - 1 ( log 5 + 2 )
Answer:
sin 3 x
1
[40]
2 2
4
-
5
- 1) ( x + 2 ) -
( 3x
tan -1
x2
- 1 2x
Answer:
Answer:
[41] sin 4x sin 2x x x + cos 2x cos 4x 32 8 4 8
[42]
Answer:
Answer:
[43] x tan- 1 x
- 1 log ( 1 + x2 )
+
+
2 3
x2
-
2x + 1
2
+
2x + 1
x
a3 x - a a sin - 1 + (x 2 a 2
x
- 2x - x 2 - 2 sin - 1
6x + 5
log
24 3
log
8
- a ) 2ax - x
- 1 -
2
tan - 1
3
( 3x
2
2
1
[46]
7(1 +
7 x2 ) 2
-
1 ( 2ax 3
x
- 1 +1
3
5(1 + x
Answer:
x2
[47] log x +
x
- a2 -
3
x
2
- a2
Answer:
Answer:
[50] 1 x3 tan -1x [ x 2 - log ( 1 + x 2 )] 3 6
[51]
Answer:
Answer:
-x e
1
-
1 17
cos ( 4x + tan
x
2
4
- 1 4 )
[54]
5 )2
Answer:
x - 1 ex x + 1
[48]
2
[49]
x + sin 2x +
sin 4x 2
Answer:
[52]
-
x sin 2x 4
-
1 x - 1 log 4 x + 1
cos 2x 8
-
1 tan - 1x 2
x sec 2 x 2
-
2
- x )2
1
-
x + 1 3
- 1) ( x + 2 )
Answer:
- 2 x tan- 1x - log 1 + x 2
2
4 2
49
2
Answer:
[53] -
1
[44] log x +
Answer:
[45]
[36] - 3
sin 5 x
tan x 2
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 8
( Answers at the end of all problems ) Answer:
Answer:
2x - 1 + log 2
[55]
2x
1 cos 2x
-
- tan- 1
2 log 2
1 x3 - 1 log 15 x3 + 4
1 2
[58]
e 2x ( sin 2x + cos 2x ) 8
( sec x tan x + log sec x + tan x
[60]
log x
5 log ( x 2 + 5 ) 6
1 log ( x 2 + 2 ) 3
-
20
( cos 4x + 2 sin 4x )
- 2
1
-
7
14
+
log 3x 2 + 2
1 7 6
tan- 1
3x 2
[62] log e x ( 3e x + 1 ) - 3 ( 2e x + 1 ) 2
- 1 -
1 10
log ( x 2 + 4 )
-
[64] 2x - 1
1 x tan - 1 10 2
π
Answer:
( cos - 1 x
- sin - 1 x ) -
2
π
x
- x2
Answer:
1 1+ log 2 1-
sin x
- tan- 1 sin x
sin x
Answer:
[66]
x3 [ 3 cos ( log x ) + sin ( log x ) ] 10
Answer:
1 2 ( a2
-
x 2 + b2 log x2 + a2 b2 )
Answer:
[68] log
Answer:
[70] - e
-
x 2
x sec 2
[71]
Answer:
x(x
- 1)2
( x + 1)
[69] - tan - 1x +
2
1 2
log ( 1 + sin x )
-
1 x3 log 3 x3 + 1
1 log ( 1 6
2 tan - 1
x 2
Answer:
[72]
1 sin x - 1 log 8 sin x + 1
Answer:
[73] -
2x
Answer:
[63] 1 log x 5
[67]
e
Answer:
Answer:
[65]
-
Answer:
Answer:
[61]
tan 3 x 3
Answer:
Answer:
[59]
tan x +
( log 2 ) 2 + 4
Answer:
[57]
[56]
- sin x ) +
2 log 1 + 2 sin x 3
-
1 4 2
log
2 sin x
- 1
2 sin x + 1
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 9
( Answers at the end of all problems ) Answer:
1 1 2 log ( 1 - cos x ) + log ( 1 + cos x ) log 1 + 2 cos x 2 3 6
[74]
Answer:
[75]
Answer:
tan x - 1 tan - 1 +
1 2
2 tan x
Answer:
2 2
-
x2 + x 2
[78] -
Answer:
- cos x )
Answer:
2 3 e
3x
4 (1 3
3
- x )2 +
[76] - 2a ( a -
2 tan x
- x )2
2 (1 5
5
[79]
- x )2
x 4
-
sin 4x 16
-
cos 4x 16
1 a4 2
- x 4 - a 2 cos - 1
x2 a 2
[82]
tan - 1
3 + 4 tan 2 x
Answer:
1 log 2
4x + 3
- 1
3 x 1 - x2 sin - 1x 2 2
[85]
4x + 3 + 1 Answer:
1
[87]
3
Answer:
Answer:
1-x 1 +x
2 x )2 + ( a 3
Answer:
[81]
[84]
+ 1
Answer:
-
tan x + 1 +
Answer:
[83] -
[86]
1
2tan x
Answer:
sin -1 ( sin x
[80]
log
-
tan x + 1
Answer:
x3 3
[77]
1
2
tan -1 (
[88]
2 tan x )
6 x2
9
- 9x + 20 + 34 log x -
+
x2
- 9x + 20
2
Answer:
[89]
x 16
Answer:
-
sin 12 x 192
-
3
sin 6x 144
[90]
Answer:
[92]
x 2 sin - 1 2 sin 2
[91]
1 a a2
-
Answer:
2 sin -1 ( sin x
[93]
- cos x )
Answer:
[95]
Answer:
x
Answer:
-
2 tan x + 1 tan- 1 3 3
2
Answer: 2
x + 1 tan- 1 3 3
1
[96]
a tan-1 tan x 2 2 b2 a - b
x 2 - 1 1 tan-1 3 3x
[94]
2x sin - 1 2 1 + x
1
Answer:
[97]
tan 2x tan- 1 2 2
1
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 10
( Answers at the end of all problems ) Answer:
cot x - 1 1 cot x tan - 1 log 2 2 2 cot x + 2 cot x
Answer:
[99]
1 5 + 4 (sin x - cos x ) log 40 5 - 4 (sinx - cos x )
[101]
[102]
3x 35 + log 9e 2x 2 36
2 (1 3 a2
- 4
Answer:
1 ( sin x 2
[100]
Answer:
[103]
2 cot x + 1
Answer:
Answer:
-
2 cot x + 1
1
[98] -
- a ) log 1 -
- cos x ) -
ax
π x log tan + 8 2 2 2 1
+ (a - 2)(1 -
1
[104]
Answer:
2 3
log
3 + ( sinx - cosx )
ax ) 2
-
+ tan-1 ( sinx + cosx )
3 - ( sinx - cosx )
Answer: 5
[105] (x + 1 ) tan-1 x -
x
[106]
4 4 x 5
3
- x +
8 4 x 3
1
Answer:
[107]
[108]
x ( tanx – secx) + log (1 + sinx )
Answer:
1
1
- 4 x 2 + 8 x 4 - 8 log x 4 + 1
Answer:
[
2 sin x -
x cox x
]
Answer:
1 1 log sec 3x - log sec 2x - log sec x 3 2
Answer:
[111]
1 (1 2
Answer:
1 xn log n xn + 1
[109]
ax ) +
1 2
1
- x- 4
Answer:
3 sinx 4
-
3x cosx x cos3x + 4 12
-
sin3x 36
Answer:
[113] x -
[110]
[112]
(
2x - a x ) sin- 1 2 a
-
1 2
ax - x 2
Answer:
3 log x + 3 + 24 log x + 4
- 30 log x + 5
[114]
ax + x 2 + a log
x +
x + a
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 11
( Answers at the end of all problems ) Answer:
1 1 log 1 + tanx + log tan 2 x 3 6
[115] Answer:
Answer:
[116] sin x - x cosx cos x + x sinx
n sin -1x 2
Answer:
3
x 1 3 - tan- 1x 2 x 2 2(1 + x )
[118] Answer:
x ( x + a ) tan - 1 a
[119]
3
-
π x 2 tan -1 3 tan 3 6 2
[120]
ax
Answer:
Answer:
[121]
[122]
cos x cos a cos - 1 cos a
- sin a log sinx +
sin 2 x
18x 1 + log 4 cosx + 3 sinx 25 25
- sin 2 a
Answer:
[123]
- 1
2 tanx
tan - 1
Answer:
2 n
[117]
1
- tanx + 1 +
Answer:
x - x
2
- 2 1 - x + cos - 1 x
[124]
Answer:
-
2 sinα
Answer:
[125] -
β - x β - α
α ) - ( β - α ) sin - 1
(β - x )(x -
Answer:
[126]
2 tanx 5 tan -1 2 5
2 + sin 2x
x 1 log + 2 4 2
[128]
2 - sin 2x
x log log x
Answer:
1 log x 4 12
- 1 -
xcot - 1 (1
1 2
- x + x 2 ) + log
x
- 2x + 2
x2 + 1
+ tan - 1 (x - 1)
[132]
x 1 ( tan - 1x ) 2 - x tan - 1x + log sec ( tan - 1x ) + ( tan - 1x ) 2 2 2
x +
Answer:
[130]
1 tanx - 2 log 5 2 tanx + 1
tan - 1 sec 2 x + 1 + cos 2 x
Answer: 2
Answer:
[135]
x log x
Answer: 2
Answer:
[133]
-
Answer:
2x 4 + 1 1 1 log x 8 + x 4 + 1 + tan - 1 4 3 3 24
Answer:
[131]
- x
Answer:
[127]
[129]
cosα + sinα cotx
4 4 x - 1 3 x + 2
[134]
Answer:
1 log x 2
- 1 - 8 log x - 2 +
[137]
1 2
log
27 log x 2
2 +
1
- tan 2 x
-
1
- tan 2 x
2
-
- 3
1 1+ log 2 1-
[136] 1 - tan 2 x 1 - tan 2 x
cosec –
1
( 1 + sin2x )
14 - INDEFINITE INTEGRATION
Page 1
( Answers at the end of all problems ) Integrate the following functions with respect to x:
(1)
2 4 x + x + 1 2 x + 1
(4)
2 log 1 2
(7)
(2)
1
-
− x2 2
1 2
(8)
2
cos x sin x
1
( 10 )
( 11 )
x 1 - x3
( 13 )
( 16 )
( 19 )
( 22 )
( 25 )
1 + cos 2x (π /2 < x < π)
1
(5)
( 14 )
- sin 2x
( - π / 4 < x < π/ 4 )
( 17 )
cos 3 x
( 20 )
sin 2 x
x 1
( 23 )
- x3
1- x 1+ x
( 26 )
3
(x
(3)
x2 + 1
sin x 1 - sin x
(6)
1 2
2
1
- 1) 2 ( x - 2 ) 2
2
2
a cos x + b sin x
x2
- 1
(9)
( 12 )
x4 + 1
x tan
- 1 x2
x4 x2 + 1
e 2x
- 1
2x
+ 1
e
1 1 + tan x
sin x cos x 3 sin 2 x
1 + sin x (0 < x < π )
( 15 )
sin x sin 2x
( 18 ) cot x
1 sin ( x - a ) sin ( x - b )
1 5 + 4 cos x + 3 sin x
1 - cos x (0 < x < π )
6
( 21 )
sin 2 x cos 4 x
1
( 24 )
1
1 2 x
( 27 )
( 30 )
- x3 - x
5
x
2
( 29 ) sin x cos 2x
- 4 cos 2 x
sin x
1 + x4
cos x
1
( 28 )
x6 + 2
3x + 1 2x
2
+ x + 1
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 2
( Answers at the end of all problems )
1
( 31 ) x
x
- a
4
( 32 )
4
1 (x
- 1) ( 4 - x )
1
( 34 ) x 2ax - x 2 ( 0 < x < 2a ) 2
( 37 ) x e
( 40 )
( 43 )
3x
x
4
x
sin - 1 1
+
( 44 )
(1 +
- 1) ( x + 2 )
x2
x +
x - 1
sin 5x sin x
( 47 )
( 50 )
2
( 52 ) x sec x tan x
3 2
2
x
( 42 )
( 45 )
( 48 )
( x - a )2 ( x > 0, a > 0 )
x 2 tan - 1x
( 53 ) e - x sin 2 2x
8 - 2x - x 2
x
2ax
cos - 1
x
2
( 58 ) e
2x
sin 3x sin x
( x + 1)
2
ex
2
( 51 ) x sin x
1
( 54 )
4
- 1
( 57 )
x2 ( x3
- 1) ( x3 + 4 )
3
( 59 ) sec x
1 - x2
x2 + 1
4
( 56 ) sec x
- x2
1 + x2 (x<0)
x
( 55 ) 2 cos x
2x - 1
( 39 ) 5 e
x2
9 x2 )2
3x + 1
( 36 )
1
x3
( 46 )
( 49 )
( 3x
( 41 ) x sinx cos 3x
+ 1
x2
2
( 35 ) cos x sin 2x
( 38 )
x2
25 - x 2
( 33 )
( 60 )
x ( 3x
2
+ 2)(x - 2)
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 3
( Answers at the end of all problems )
( 61 )
( 64 )
( 67 )
x3 ( x2 + 2 ) ( x2 + 5 )
cos -1 x
- sin - 1 x x + sin -1 x
cos -1
x ( x 2 + a2 ) ( x 2 + b2 )
-
1 - sin x e ( 70 ) 1 + cos x (0 < x < π / 2 )
( 73 )
( 76 )
( 85 )
2
1 cos x ( 1 + 2 sin x )
x a - x
tan x 3 + 4 tan 2 x
1 + x2 1
( 88 )
( 65 )
( 68 )
1 6e
2x
+ 5e
x
+ 1
sin x cos x
1
( 63 )
( 66 ) x
x 2 + 4x - 1 x3 - x
- 1) ( x2 + 4 )
(x
( 69 )
2
cos ( log x )
x2 ( x 2 + 1) ( x 2 + 2)
x
( 79 ) sin x cos x ( sin 2x + cos 2x )
( 82 )
( 62 )
- x2 6x + 7
(x - 5)(x - 4)
1
( 71 ) x(x
( 74 )
( 77 )
( 80 )
3
( 72 )
+ 1)
1 sin x + sin 2x
x4 + x2 + 1 x2 + x + 1
sin x sin 4 x
( 75 )
tan x
sin x + cos x
( 81 )
sin 2x
1
( 83 ) ( ex +
3 2x 2 e )
1
( 86 )
(1 + x ) 1
4
- x2
( 84 )
( 87 )
2
( 89 ) sin 3x cos 3x
1- x
( 78 ) ( 1 + x )
( 90 )
x
a2 - x2 a2 + x2
1 ( 2x + 1 ) 4x + 3
1 cos 2x + 3 sin 2 x
sec x + 1 0 < x < π, x ≠ π / 2
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 4
( Answers at the end of all problems )
1
( 91 )
a - b 2 cos 2 x (a > b)
1
( 94 )
( 1 + x ) 1 + 2x
( 97 )
( 100 )
( 103 )
( 106 )
- x
( 115 )
( 118 )
cot x
2
2
(x + 4)
2
x + 1
1 sin 4 x + cos 4 x
1 sec x + cosec x
1 x ( xn + 1)
1+
x
1+
1 4 x
x a - x
tan- 1
tan x + tan 3 x 1 + tan 3 x
1 2
2 2
x (1 + x )
( 93 )
tan x 1 + tan x + tan 2 x
x
( 95 )
( 109 ) tanx tan2x tan 3x
( 112 )
tan x +
( 92 )
2
( 98 )
( 101 )
( 96 )
( 99 )
cot x
4e x + 6e - x 9e x - 4e - x
( 102 )
( 104 )
1 sin x + secx
( 105 )
( 107 )
x 1 + sinx
( 108 )
( 110 )
( 113 )
( 116 )
( 119 )
1 x3
x4 - 1
x ( x + 1) ( x + 2 ) ( x + 3)( x + 4)(x + 5)
x2 ( x sinx + cosx ) 2
sin - 1
x a + x
x2 x4
1
+
7x 2 + 1
+
sin x + cosx 9 + 16 sin2x
a 1-
x ax
tan-1 x
sin x
3
( 111 ) x sin x
( 114 )
( 117 )
( 120 )
a + x x xn -
2
1 - xn
1 5 - 2 cosx - 2 3 sinx
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 5
( Answers at the end of all problems )
sin ( x - a ) sin ( x + a )
( 121 )
( 122 )
1
( 124 )
sin 3 x sin ( x +
α)
x - α β - x
( 126 )
( 128 ) log log x +
4
1 + tan x
( 130 ) 1 ( sinx - 2cosx ) ( 2sinx + cosx )
( 131 )
( 133 ) x ( tan – 1x ) 2
( 134 )
sin x
( 136 )
( 123 )
( 125 )
1
( 127 )
3 + 2 tanx 4 + 3 tanx
1
cot -1 ( 1 - x + x 2 )
x
1+
x
sin 2 x 4 + cos 2 x
x7
( 129 )
( log x ) 2
-
1
x 12 - 1
( 132 ) sin 3 x ( 1 + cos 2 x ) 1 + cos 2 x + cos 4 x
1 4
(x
- 1 )3 ( x + 2 )5
- cosx 2
2
( sinx + cosx ) sinx cosx + sin x cos x
( 137 )
( 135 )
x3 (x
- 1) ( x - 2 ) ( x - 3 )
cos 2x sin x
Answers ( Add constant C to all answers )
Answer:
[1]
Answer:
x3 -1 + tan x 3
Answer:
[4]
sin - 1
x log 2
[2]
x5 5
Answer:
-
x3 + x + tan - 1 x 3
[3]
x3 - x + tan- 1 x 3
Answer:
Answer:
[ 5 ] sec x + tan x - x
[ 6 ] log ( e x + e – x )
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 6
( Answers at the end of all problems ) Answer:
Answer:
[ 7 ] tan x - cot x
[8]
Answer:
Answer:
1 - x3 - 1
1 log [ 10 ] 3
[ 13 ]
1 b tan- 1 tan x ab a
[ 11 ]
1 - x3 + 1
Answer:
Answer:
Answer: 1
log
2 2
x2 x
Answer:
- 2 sin x
1
[ 14 ]
Answer:
4
x 1 + log I cos x + sin x I 2 2
[9]
2
- 2x + 1 +
1
[ 12 ]
14
2x + 1
log I 3 sin 2 x
- 4 cos 2 x I
Answer:
( tan - 1 x 2 ) 2
[ 15 ] - 2 2
Answer:
sin (
π 4
x
-
2
1
) -
2
log tan (
π
-
8
x 4
)
Answer: 5
4 ( sin x ) 2 5
[ 18 ] -
[ 16 ] sin x + cos x
[ 17 ]
Answer:
Answer:
Answer:
[ 20 ]
[ 21 ]
1
[ 19 ] -
sin x
- sin x
Answer:
Answer:
sin -
2
[ 22 ]
1 sin ( x - a ) log sin ( a - b ) sin ( x - b )
3
3 1 2 x
1 192
( 12x
[ 23 ] tan
2 x 2
[ 24 ]
1 2 x2
1
[25]
[26]
[27]
Answer:
x - 2 x - 1
[29]
Answer:
2
x sec - 1 a
2
sin 2x x sin 4x 4 4 16
2
[32] 2 sin - 1
x
2
2
5 - x - 5 log
5 - x +
5
5 - x -
5
Answer:
Answer:
1 2a
4
+ 2 2 cos
Answer:
[28] 2
[31]
x
1
+ 3 Answer:
2 log tan
1
- 3 x 3 + 6 x 6 - 6 log ( 1 + x 6 )
Answer:
1 - x2
- 3 sin 4x + 3 sin 2x - sin 6x )
Answer:
Answer:
sin -1 x +
cot 5 x cot 3 x + - cot x - x 5 3
[30]
3 1 4x + 1 log 2x 2 + x + 1 + tan- 1 4 2 7 7 Answer:
x - 1 3
[33] -
25
- x2 x
- sin - 1
x 5
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 7
( Answers at the end of all problems ) Answer:
[34] -
Answer:
1
- x
2a
a
4
[35]
3
a
Answer:
Answer:
x2 2x 2 3x [37] + e 9 27 3
[38] 6x + 5 12
Answer:
Answer:
[39]
5 x e 2x - 1 ( log 5 + 2 )
Answer:
sin 3 x
1
[40]
2 2
4
-
5
- 1) ( x + 2 ) -
( 3x
tan -1
x2
- 1 2x
Answer:
Answer:
[41] sin 4x sin 2x x x + cos 2x cos 4x 32 8 4 8
[42]
Answer:
Answer:
[43] x tan- 1 x
- 1 log ( 1 + x2 )
+
+
2 3
x2
-
2x + 1
2
+
2x + 1
x
a3 x - a a sin - 1 + (x 2 a 2
x
- 2x - x 2 - 2 sin - 1
6x + 5
log
24 3
log
8
- a ) 2ax - x
- 1 -
2
tan - 1
3
( 3x
2
2
1
[46]
7(1 +
7 x2 ) 2
-
1 ( 2ax 3
x
- 1 +1
3
5(1 + x
Answer:
x2
[47] log x +
x
- a2 -
3
x
2
- a2
Answer:
Answer:
[50] 1 x3 tan -1x [ x 2 - log ( 1 + x 2 )] 3 6
[51]
Answer:
Answer:
-x e
1
-
1 17
cos ( 4x + tan
x
2
4
- 1 4 )
[54]
5 )2
Answer:
x - 1 ex x + 1
[48]
2
[49]
x + sin 2x +
sin 4x 2
Answer:
[52]
-
x sin 2x 4
-
1 x - 1 log 4 x + 1
cos 2x 8
-
1 tan - 1x 2
x sec 2 x 2
-
2
- x )2
1
-
x + 1 3
- 1) ( x + 2 )
Answer:
- 2 x tan- 1x - log 1 + x 2
2
4 2
49
2
Answer:
[53] -
1
[44] log x +
Answer:
[45]
[36] - 3
sin 5 x
tan x 2
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 8
( Answers at the end of all problems ) Answer:
Answer:
2x - 1 + log 2
[55]
2x
1 cos 2x
-
- tan- 1
2 log 2
1 x3 - 1 log 15 x3 + 4
1 2
[58]
e 2x ( sin 2x + cos 2x ) 8
( sec x tan x + log sec x + tan x
[60]
log x
5 log ( x 2 + 5 ) 6
1 log ( x 2 + 2 ) 3
-
20
( cos 4x + 2 sin 4x )
- 2
1
-
7
14
+
log 3x 2 + 2
1 7 6
tan- 1
3x 2
[62] log e x ( 3e x + 1 ) - 3 ( 2e x + 1 ) 2
- 1 -
1 10
log ( x 2 + 4 )
-
[64] 2x - 1
1 x tan - 1 10 2
π
Answer:
( cos - 1 x
- sin - 1 x ) -
2
π
x
- x2
Answer:
1 1+ log 2 1-
sin x
- tan- 1 sin x
sin x
Answer:
[66]
x3 [ 3 cos ( log x ) + sin ( log x ) ] 10
Answer:
1 2 ( a2
-
x 2 + b2 log x2 + a2 b2 )
Answer:
[68] log
Answer:
[70] - e
-
x 2
x sec 2
[71]
Answer:
x(x
- 1)2
( x + 1)
[69] - tan - 1x +
2
1 2
log ( 1 + sin x )
-
1 x3 log 3 x3 + 1
1 log ( 1 6
2 tan - 1
x 2
Answer:
[72]
1 sin x - 1 log 8 sin x + 1
Answer:
[73] -
2x
Answer:
[63] 1 log x 5
[67]
e
Answer:
Answer:
[65]
-
Answer:
Answer:
[61]
tan 3 x 3
Answer:
Answer:
[59]
tan x +
( log 2 ) 2 + 4
Answer:
[57]
[56]
- sin x ) +
2 log 1 + 2 sin x 3
-
1 4 2
log
2 sin x
- 1
2 sin x + 1
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 9
( Answers at the end of all problems ) Answer:
1 1 2 log ( 1 - cos x ) + log ( 1 + cos x ) log 1 + 2 cos x 2 3 6
[74]
Answer:
[75]
Answer:
tan x - 1 tan - 1 +
1 2
2 tan x
Answer:
2 2
-
x2 + x 2
[78] -
Answer:
- cos x )
Answer:
2 3 e
3x
4 (1 3
3
- x )2 +
[76] - 2a ( a -
2 tan x
- x )2
2 (1 5
5
[79]
- x )2
x 4
-
sin 4x 16
-
cos 4x 16
1 a4 2
- x 4 - a 2 cos - 1
x2 a 2
[82]
tan - 1
3 + 4 tan 2 x
Answer:
1 log 2
4x + 3
- 1
3 x 1 - x2 sin - 1x 2 2
[85]
4x + 3 + 1 Answer:
1
[87]
3
Answer:
Answer:
1-x 1 +x
2 x )2 + ( a 3
Answer:
[81]
[84]
+ 1
Answer:
-
tan x + 1 +
Answer:
[83] -
[86]
1
2tan x
Answer:
sin -1 ( sin x
[80]
log
-
tan x + 1
Answer:
x3 3
[77]
1
2
tan -1 (
[88]
2 tan x )
6 x2
9
- 9x + 20 + 34 log x -
+
x2
- 9x + 20
2
Answer:
[89]
x 16
Answer:
-
sin 12 x 192
-
3
sin 6x 144
[90]
Answer:
[92]
x 2 sin - 1 2 sin 2
[91]
1 a a2
-
Answer:
2 sin -1 ( sin x
[93]
- cos x )
Answer:
[95]
Answer:
x
Answer:
-
2 tan x + 1 tan- 1 3 3
2
Answer: 2
x + 1 tan- 1 3 3
1
[96]
a tan-1 tan x 2 2 b2 a - b
x 2 - 1 1 tan-1 3 3x
[94]
2x sin - 1 2 1 + x
1
Answer:
[97]
tan 2x tan- 1 2 2
1
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 10
( Answers at the end of all problems ) Answer:
cot x - 1 1 cot x tan - 1 log 2 2 2 cot x + 2 cot x
Answer:
[99]
1 5 + 4 (sin x - cos x ) log 40 5 - 4 (sinx - cos x )
[101]
[102]
3x 35 + log 9e 2x 2 36
2 (1 3 a2
- 4
Answer:
1 ( sin x 2
[100]
Answer:
[103]
2 cot x + 1
Answer:
Answer:
-
2 cot x + 1
1
[98] -
- a ) log 1 -
- cos x ) -
ax
π x log tan + 8 2 2 2 1
+ (a - 2)(1 -
1
[104]
Answer:
2 3
log
3 + ( sinx - cosx )
ax ) 2
-
+ tan-1 ( sinx + cosx )
3 - ( sinx - cosx )
Answer: 5
[105] (x + 1 ) tan-1 x -
x
[106]
4 4 x 5
3
- x +
8 4 x 3
1
Answer:
[107]
[108]
x ( tanx – secx) + log (1 + sinx )
Answer:
1
1
- 4 x 2 + 8 x 4 - 8 log x 4 + 1
Answer:
[
2 sin x -
x cox x
]
Answer:
1 1 log sec 3x - log sec 2x - log sec x 3 2
Answer:
[111]
1 (1 2
Answer:
1 xn log n xn + 1
[109]
ax ) +
1 2
1
- x- 4
Answer:
3 sinx 4
-
3x cosx x cos3x + 4 12
-
sin3x 36
Answer:
[113] x -
[110]
[112]
(
2x - a x ) sin- 1 2 a
-
1 2
ax - x 2
Answer:
3 log x + 3 + 24 log x + 4
- 30 log x + 5
[114]
ax + x 2 + a log
x +
x + a
PROBLEMS
14 - INDEFINITE INTEGRATION
Page 11
( Answers at the end of all problems ) Answer:
1 1 log 1 + tanx + log tan 2 x 3 6
[115] Answer:
Answer:
[116] sin x - x cosx cos x + x sinx
n sin -1x 2
Answer:
3
x 1 3 - tan- 1x 2 x 2 2(1 + x )
[118] Answer:
x ( x + a ) tan - 1 a
[119]
3
-
π x 2 tan -1 3 tan 3 6 2
[120]
ax
Answer:
Answer:
[121]
[122]
cos x cos a cos - 1 cos a
- sin a log sinx +
sin 2 x
18x 1 + log 4 cosx + 3 sinx 25 25
- sin 2 a
Answer:
[123]
- 1
2 tanx
tan - 1
Answer:
2 n
[117]
1
- tanx + 1 +
Answer:
x - x
2
- 2 1 - x + cos - 1 x
[124]
Answer:
-
2 sinα
Answer:
[125] -
β - x β - α
α ) - ( β - α ) sin - 1
(β - x )(x -
Answer:
[126]
2 tanx 5 tan -1 2 5
2 + sin 2x
x 1 log + 2 4 2
[128]
2 - sin 2x
x log log x
Answer:
1 log x 4 12
- 1 -
xcot - 1 (1
1 2
- x + x 2 ) + log
x
- 2x + 2
x2 + 1
+ tan - 1 (x - 1)
[132]
x 1 ( tan - 1x ) 2 - x tan - 1x + log sec ( tan - 1x ) + ( tan - 1x ) 2 2 2
x +
Answer:
[130]
1 tanx - 2 log 5 2 tanx + 1
tan - 1 sec 2 x + 1 + cos 2 x
Answer: 2
Answer:
[135]
x log x
Answer: 2
Answer:
[133]
-
Answer:
2x 4 + 1 1 1 log x 8 + x 4 + 1 + tan - 1 4 3 3 24
Answer:
[131]
- x
Answer:
[127]
[129]
cosα + sinα cotx
4 4 x - 1 3 x + 2
[134]
Answer:
1 log x 2
- 1 - 8 log x - 2 +
[137]
1 2
log
27 log x 2
2 +
1
- tan 2 x
-
1
- tan 2 x
2
-
- 3
1 1+ log 2 1-
[136] 1 - tan 2 x 1 - tan 2 x
cosec –
1
( 1 + sin2x )
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