Page 710 - Handbook Of Integral Equations


1 px a
 arctan e if ab >0,
 √
dx p ab b

8. = px √
ae px + be –px  1 b + e –ab

√ ln √ if ab <0.

2p –ab b – e px –ab
√
 √
1 a + be px – a

 √ ln √
 if a >0,
dx  p a a + be px + a
√
9. √ = √
a + be px  2 a + be px
 √ arctan √ if a <0.

p –a –a

2.4. Integrals Containing Hyperbolic Functions
Integrals containing cosh x.

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

2. x cosh xdx = x sinh x – cosh x.

2
2
3. x cosh xdx =(x + 2) sinh x – 2x cosh x.
n 2k 2k–1
x x
2n
4. x cosh xdx =(2n)! sinh x – cosh x .
(2k)! (2k – 1)!
k=1
n
x 2k+1 x 2k
5. x 2n+1 cosh xdx =(2n + 1)! sinh x – cosh x .
(2k + 1)! (2k)!
k=0

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

2
1
7. cosh xdx = x + 1 sinh 2x.
2 4

3
3
8. cosh xdx = sinh x + 1 sinh x.
3
n–1
x 1 sinh[2(n – k)x]
9. cosh 2n xdx = C n + C k , n =1, 2, ...
2n 2n 2n–1 2n
2 2 2(n – k)
k=0
n n
1 k sinh[(2n – 2k +1)x] k sinh x
2k+1
2n+1
10. cosh xdx = C 2n+1 = C n , n =1, 2, ...
2 2n 2n – 2k +1 2k +1
k=0 k=0

p 1 p–1 p – 1 p–2
11. cosh xdx = sinh x cosh x + cosh xdx.
p p

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

13. = arctan e .
cosh ax a
n–1
dx sinh x 1 2 (n – 1)(n – 2) ... (n – k) 1
k
14. 2n = 2n–1 + 2n–2k–1 ,
cosh x 2n – 1 cosh x (2n – 3)(2n – 5) ... (2n – 2k – 1) cosh x
k=1
n =1, 2, ...
n–1
dx sinh x 1 (2n – 1)(2n – 3) ... (2n – 2k +1) 1

15. 2n+1 = 2n + 2n–2k
k
cosh x 2n cosh x 2 (n – 1)(n – 2) ... (n – k) cosh x
k=1
(2n – 1)!!
+ arctan sinh x, n =1, 2, ...
(2n)!!
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 694