Page 717 - Handbook Of Integral Equations

p–1 q+1
sin x cos x p – 1 p–2 q+2
q
p
63. sin x cos xdx = – + sin x cos xdx.
q +1 q +1
sin x cos x q – 1 p+2 q–2
p+1 q–1
q
p
64. sin x cos xdx = + sin x cos xdx.
p +1 p +1
Integrals containing tan x and cot x.

65. tan xdx = – ln |cos x|.

2
66. tan xdx = tan x – x.

2
3
67. tan xdx = 1 tan x +ln |cos x|.
2
n
k 2n–2k+1
2n n (–1) (tan x)
68. tan xdx =(–1) x – , n =1, 2, ...
2n – 2k +1
k=1
n
k
(–1) (tan x) 2n–2k+2
69. tan 2n+1 xdx =(–1) n+1 ln |cos x| – , n =1, 2, ...
2n – 2k +2
k=1
dx 1

70. = ax + b ln |a cos x + b sin x| .
2
a + b tan x a + b 2

tan xdx 1 a

71. √ = √ arccos 1 – cos x , b > a, b >0.
2
a + b tan x b – a b

72. cot xdx =ln |sin x|.

2
73. cot xdx = – cot x – x.

3
1
2
74. cot xdx = – cot x – ln |sin x|.
2
n
k 2n–2k+1
(–1) (cot x)
2n
n
75. cot xdx =(–1) x + , n =1, 2, ...
2n – 2k +1
k=1
n
k
(–1) (cot x) 2n–2k+2
n
76. cot 2n+1 xdx =(–1) ln |sin x| + , n =1, 2, ...
2n – 2k +2
k=1

dx 1
77. = ax – b ln |a sin x + b cos x| .
2
a + b cot x a + b 2
2.7. Integrals Containing Inverse Trigonometric
Functions
x x √

1. arcsin dx = x arcsin + a – x .
2
2
a a

x x x
2 2 √
2
2
2. arcsin dx = x arcsin – 2x +2 a – x arcsin .
a a a
x 1 2 2 x x
√
2
2
3. x arcsin dx = (2x – a ) arcsin + a – x .
a 4 a 4
x x x 1 2 2
3 √
2
2
2
4. x arcsin dx = arcsin + (x +2a ) a – x .
a 3 a 9
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 701