Page 711 - Handbook Of Integral Equations

sign x b + a cosh x 2 2

 – √ arcsin if a < b ,

dx  b – a a + b cosh x
2 2
16. = √ 2 2
a + b cosh x  1 a + b + a – b tanh(x/2) 2 2
 √ ln √ if a > b .

2
2
2
a – b 2 a + b – a – b tanh(x/2)
Integrals containing sinh x.

1
17. sinh(a + bx) dx = cosh(a + bx).
b

18. x sinh xdx = x cosh x – sinh x.

2
2
19. x sinh xdx =(x + 2) cosh x – 2x sinh x.
n n
x 2k x 2k–1
20. x 2n sinh xdx =(2n)! cosh x – sinh x .
(2k)! (2k – 1)!
k=0 k=1
n
x 2k+1 x 2k
21. x 2n+1 sinh xdx =(2n + 1)! cosh x – sinh x .
(2k + 1)! (2k)!
k=0

p
p
22. x sinh xdx = x cosh x – px p–1 sinh x + p(p – 1) x p–2 sinh xdx.

1
2
23. sinh xdx = – x + 1 sinh 2x.
2 4

3
3
24. sinh xdx = – cosh x + 1 cosh x.
3
n–1

x 1 k k sinh[2(n – k)x]
2n
n
n
25. sinh xdx =(–1) C + (–1) C , n =1, 2, ...
2n 2n 2n–1 2n
2 2 2(n – k)
k=0
n n
1 k k cosh[(2n – 2k +1)x] n+k k cosh x
2k+1
2n+1
26. sinh xdx = (–1) C 2n+1 = (–1) C n ,
2 2n 2n – 2k +1 2k +1
k=0 k=0
n =1, 2, ...
1 p – 1
p
27. sinh xdx = sinh p–1 x cosh x – sinh p–2 xdx.
p p

1
28. sinh ax sinh bx dx = a cosh ax sinh bx – b cosh bx sinh ax .
2
a – b 2
dx 1

29. = ln tanh ax .

sinh ax a 2

dx cosh x 1
30. 2n = – 2n–1
sinh x 2n – 1 sinh x
n–1 k
k–1 2 (n – 1)(n – 2) ... (n – k) 1
+ (–1) , n =1, 2, ...
(2n – 3)(2n – 5) ... (2n – 2k – 1) sinh 2n–2k–1 x
k=1
n–1
dx cosh x 1 k–1 (2n–1)(2n–3) ... (2n–2k+1) 1

31. 2n+1 = – 2n + (–1) k 2n–2k
sinh x 2n sinh x 2 (n–1)(n–2) ... (n–k) sinh x
k=1
(2n – 1)!! x
n
+(–1) ln tanh , n =1, 2, ...
(2n)!! 2
√
dx 1 a tanh(x/2) – b + a + b
2 2
32. = √ ln √ .
a + b sinh x a + b 2 a tanh(x/2) – b – a + b 2
2
2
√
2 2
Ax + B sinh x B Ab – Ba a tanh(x/2) – b + a + b
33. dx = x + √ ln √ .
a + b sinh x b b a + b 2 a tanh(x/2) – b – a + b 2
2
2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 695