Page 714 - Handbook Of Integral Equations

3
3
8. cos xdx = sin x – 1 sin x.
3
n–1
1 n 1 k sin[(2n – 2k)x]

2n
9. cos xdx = C x + C 2n .
2n
2 2n 2 2n–1 2n – 2k
k=0
n

1 k sin[(2n – 2k +1)x]
2n+1
10. cos xdx = C 2n+1 .
2 2n 2n – 2k +1
k=0
dx x

11. =ln tan + π .

cos x 2 4
dx

12. = tan x.
2
cos x
dx sin x 1 x

13. = + ln tan + π .

3
2
cos x 2 cos x 2 2 4

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

15. =
cos 2n x (2n – 1)(2n – 3) ... (2n – 2k +3) (2n – 2k + 1)(2n – 2k) cos 2n–2k+1 x
k=0
2 n–1 (n – 1)!
+ x tan x +ln |cos x| .
(2n – 1)!!

sin (b – a)x sin (b + a)x

16. cos ax cos bx dx = + , a ≠ ±b.
2(b – a) 2(b + a)

2 (a – b) tan(x/2) 2 2

 √ arctan √ if a > b ,

dx a – b a – b
 2 2 2 2
17. = √
2
2
a + b cos x  1 b – a +(b – a) tan(x/2) 2 2

 √ ln √ if b > a .

b – a b – a – (b – a) tan(x/2)
 2 2 2 2

dx b sin x a dx
18. = – .
2
2
2
(a + b cos x) 2 (b – a )(a + b cos x) b – a 2 a + b cos x
dx 1 a tan x

19. = √ arctan √ .
a + b cos x a a + b 2 a + b 2
2
2
2
2
2
1 a tan x

2
2
 √ arctan √ if a > b ,

 2 2 2 2
dx a a – b a – b
20. = √
2
2
2
2
a – b cos x  1 b – a – a tan x
2
√ ln √ if b > a .
 2 2


2
2
2
2a b – a 2 b – a + a tan x

b a
21. e ax cos bx dx = e ax sin bx + cos bx .
2
2
a + b 2 a + b 2
e ax 2
2
2
22. e ax cos xdx = a cos x + 2 sin x cos x + .
2
a +4 a
ax n–1
e cos x n(n – 1) ax n–2
n
ax
23. e cos xdx = (a cos x + n sin x)+ e cos xdx.
2
a + n 2 a + n 2
2
Integrals containing sin x. Notation: n =1, 2, ...
1

24. sin(a + bx) dx = – cos(a + bx).
b

25. x sin xdx = sin x – x cos x.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 698