Page 715 - Handbook Of Integral Equations

2
2
26. x sin xdx =2x sin x – (x – 2) cos x.

3
2
3
27. x sin xdx =(3x – 6) sin x – (x – 6x) cos x.
n n–1
2n–2k 2n–2k–1
k+1 x k x
2n
28. x sin xdx =(2n)! (–1) cos x + (–1) sin x .
(2n – 2k)! (2n – 2k – 1)!
k=0 k=0

n x 2n–2k+1 x 2n–2k

29. x 2n+1 sin xdx =(2n + 1)! (–1) k+1 cos x +(–1) k sin x .
(2n – 2k + 1)! (2n – 2k)!
k=0

p
p
30. x sin xdx = –x cos x + px p–1 sin x – p(p – 1) x p–2 sin xdx.

1
2
31. sin xdx = x – 1 sin 2x.
2 4

2
1
1
2
32. x sin xdx = x – x sin 2x – 1 cos 2x.
4 4 8

3
3
33. sin xdx = – cos x + 1 cos x.
3
n–1
1 (–1) n sin[(2n – 2k)x]
k
k
n
34. sin 2n xdx = C x + (–1) C 2n ,
2n
2 2n 2 2n–1 2n – 2k
k=0
m!
k
where C m = are binomial coefficients (0! = 1).
k!(m – k)!
n
1 n+k+1 k cos[(2n – 2k +1)x]

2n+1
35. sin xdx = (–1) C 2n+1 .
2 2n 2n – 2k +1
k=0
dx

36. =ln tan x .

sin x 2
dx

37. = – cot x.
2
sin x
dx cos x 1

38. = – + ln tan x .

2
3
sin x 2 sin x 2 2
dx cos x n – 2 dx

39. = – + , n >1.
n
sin x (n – 1) sin n–1 x n – 1 sin n–2 x
n–1
xdx (2n – 2)(2n – 4) ... (2n – 2k +2) sin x +(2n – 2k)x cos x

40. = –
sin 2n x (2n – 1)(2n – 3) ... (2n – 2k +3) (2n – 2k + 1)(2n – 2k) sin 2n–2k+1 x
k=0
2 n–1 (n – 1)!
+ ln |sin x| – x cot x .
(2n – 1)!!
sin[(b – a)x] sin[(b + a)x]

41. sin ax sin bx dx = – , a ≠ ±b.
2(b – a) 2(b + a)
2 b + a tan x/2 2 2


 √ arctan √ if a > b ,

dx a – b a – b
 2 2 2 2
42. = √
2
a + b sin x  1 b – b – a + a tan x/2 2 2
2
 √ ln √ if b > a .

 2 2 2 2
b – a b + b – a + a tan x/2

dx b cos x a dx
43. = + .
2
2
2
(a + b sin x) 2 (a – b )(a + b sin x) a – b 2 a + b sin x
√
2 2
dx 1 a + b tan x
44. = √ arctan .
2
2
2
a + b sin x a a + b 2 a
2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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