Page 716 - Handbook Of Integral Equations

√
 2 2
1 a – b tan x
 2 2
 √ arctan if a > b ,

dx  a a – b 2 a
2

45. = √
2
2
2
a – b sin x  1 2 2
 b – a tan x + a 2 2
 √ ln √ if b > a .


2
2
2a b – a 2 b – a tan x – a
2

sin xdx 1 k cos x
46. √ = – arcsin √ .
2
2
1+ k sin x k 1+ k 2
√
sin xdx 1
2
47. √ = – ln k cos x + 1 – k sin x .
2

2
2
1 – k sin x k
√ cos x √ 1+ k k cos x
2
2
2
2
2
48. sin x 1+ k sin xdx = – 1+ k sin x – arcsin √ .
2 2k 1+ k 2
2
√ cos x √ 1 – k √
2
2
2
2
2
49. sin x 1 – k sin xdx = – 1 – k sin x – ln k cos x + 1 – k sin x .

2
2 2k

a b
50. e ax sin bx dx = e ax sin bx – cos bx .
2
a + b 2 a + b 2
2
e ax 2
2
2
51. e ax sin xdx = a sin x – 2 sin x cos x + .
2
a +4 a
e sin x n(n – 1) ax n–2
ax n–1
ax
n
52. e sin xdx = (a sin x – n cos x)+ e sin xdx.
2
a + n 2 a + n 2
2
Integrals containing sin x and cos x.

cos[(a + b)x] cos (a – b)x

53. sin ax cos bx dx = – – , a ≠ ±b.
2(a + b) 2(a – b)
dx 1 c

54. = arctan tan ax .
2
2
2
2
b cos ax + c sin ax abc b

dx 1
55. = c tan ax + b .
2
2
2
2
b cos ax – c sin ax 2abc ln c tan ax – b
n+m–1
dx k tan x
2k–2m+1
56. = C n+m–1 , n, m =1, 2, ...
cos 2n x sin 2m x 2k – 2m +1
k=0
n+m
dx m k tan x
2k–2m
57. = C n+m ln |tan x| + C n+m , n, m =1, 2, ...
cos 2n+1 x sin 2m+1 x 2k – 2m
k=0
Reduction formulas. The parameters p and q below can assume any values, except for those at
which the denominators on the right-hand side vanish.
sin p–1 x cos q+1 x p – 1
p
q
q
58. sin x cos xdx = – + sin p–2 x cos xdx.
p + q p + q
sin x cos x q – 1 p q–2
p+1 q–1
p
q
59. sin x cos xdx = + sin x cos xdx.
p + q p + q
p–1 q–1
p q sin x cos x 2 q – 1
60. sin x cos xdx = sin x –
p + q p + q – 2
(p – 1)(q – 1) p–2 q–2
+ sin x cos xdx.
(p + q)(p + q – 2)
sin p+1 x cos q+1 x p + q +2
q
p
q
61. sin x cos xdx = + sin p+2 x cos xdx.
p +1 p +1
sin x cos x p + q +2 p q+2
p+1 q+1
q
p
62. sin x cos xdx = – + sin x cos xdx.
q +1 q +1
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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