Page 712 - Handbook Of Integral Equations

Integrals containing tanh x or coth x.

34. tanh xdx = ln cosh x.

2
35. tanh xdx = x – tanh x.

2
3
1
36. tanh xdx = – tanh x + ln cosh x.
2
n
tanh 2n–2k+1 x
37. tanh 2n xdx = x – , n =1, 2, ...
2n – 2k +1
k=1
n n
k
(–1) C k tanh 2n–2k+2 x
38. tanh 2n+1 xdx = ln cosh x – 2k n = ln cosh x – , n =1, 2, ...
2k cosh x 2n – 2k +2
k=1 k=1
1
p
39. tanh xdx = – tanh p–1 x + tanh p–2 xdx.
p – 1

40. coth xdx =ln |sinh x|.

2
41. coth xdx = x – coth x.

2
1
3
42. coth xdx = – coth x +ln |sinh x|.
2
n
2n–2k+1
coth x
2n
43. coth xdx = x – , n =1, 2, ...
2n – 2k +1
k=1
n n
k 2n–2k+2
C n coth x
2n+1
44. coth xdx =ln |sinh x|– 2k =ln |sinh x|– , n =1, 2, ...
2k sinh x 2n – 2k +2
k=1 k=1

1 p–1 p–2
p
45. coth xdx = – coth x + coth xdx.
p – 1
2.5. Integrals Containing Logarithmic Functions

1. ln ax dx = x ln ax – x.

2
2
1
1
2. x ln xdx = x ln x – x .
2 4
1 1

p+1 p+1
 x ln ax – x if p ≠ –1,
p
3. x ln ax dx = p +1 (p +1) 2
2
 1 ln ax if p = –1.
2

2
2
4. (ln x) dx = x(ln x) – 2x ln x +2x.

1
2
2
2
1
2
2
1
5. x(ln x) dx = x (ln x) – x ln x + x .
2 2 4
 p+1 p+1 p+1
 x 2x 2x
2
 (ln x) – ln x + if p ≠ –1,
2
p
6. x (ln x) dx = p +1 (p +1) 2 (p +1) 3
3
 1 ln x if p = –1.

3
n
x k n–k

n
7. (ln x) dx = (–1) (n +1)n... (n – k + 1)(ln x) , n =1, 2, ...
n +1
k=0
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 696