Page 704 - Handbook Of Integral Equations

xdx b a
15. = – ln a + x .
(a + x)(b + x) 2 (a – b)(b + x) (a – b) 2 b + x
x dx b a b – 2ab
2 2 2 2
16. = + ln |a + x| + ln |b + x|.
(a + x)(b + x) 2 (b – a)(b + x) (a – b) 2 (b – a) 2

dx 1 1 1 2
17. = – + + a + x .
(a + x) (b + x) 2 (a – b) 2 a + x b + x (a – b) 3 ln b + x
2

xdx 1 a b a + b
18. = + + ln a + x .
2
(a + x) (b + x) 2 (a – b) 2 a + x b + x (a – b) 3 b + x
x dx 1 a b 2 2ab
2 2
19. = – + + a + x .
2
(a + x) (b + x) 2 (a – b) 2 a + x b + x (a – b) 3 ln b + x
2
2
Integrals containing a + x .

dx 1 x
20. = arctan .
a + x 2 a a
2
dx x 1 x

21. = + arctan .
2
2 2
2
2
2
(a + x ) 2a (a + x ) 2a 3 a
dx x 3x 3 x

22. = + + arctan .
2
2
4
2
2
2
2 3
2 2
(a + x ) 4a (a + x ) 8a (a + x ) 8a 5 a
dx x 2n – 1 dx

23. = + ; n =1, 2, ...
2 n+1
2
2 n
2
2
2 n
2
(a + x ) 2na (a + x ) 2na 2 (a + x )
xdx 1 2 2

24. = ln(a + x ).
2
a + x 2 2
xdx 1

25. = – .
2 2
2
2
2
(a + x ) 2(a + x )
xdx 1

26. = – .
2 3
2
2 2
2
(a + x ) 4(a + x )
xdx 1

27. = – ; n =1, 2, ...
2 n
2
2 n+1
2
(a + x ) 2n(a + x )
2
x dx x
28. = x – a arctan .
2
a + x 2 a
2
x dx x 1 x
29. = – + arctan .
2
2
2 2
2
(a + x ) 2(a + x ) 2a a
2
x dx x x 1 x
30. = – + + arctan .
2
2 3
2
(a + x ) 4(a + x ) 8a (a + x ) 8a 3 a
2
2
2 2
2
x dx x 1 dx
2
31. = – + ; n =1, 2, ...
2
2 n
2
2 n+1
2
2 n
(a + x ) 2n(a + x ) 2n (a + x )
3 2 2
x dx x a 2 2
32. = – ln(a + x ).
a + x 2 2 2
2
x dx a 1 2 2
3 2
33. = + ln(a + x ).
2
2
2 2
2
(a + x ) 2(a + x ) 2
3 2
x dx 1 a
34. = – + ; n =2, 3, ...
2 n+1
2 n
2 n–1
2
2
2
(a + x ) 2(n – 1)(a + x ) 2n(a + x )
dx 1 x
2
35. = ln .
2
x(a + x ) 2a 2 a + x 2
2
2
dx 1 1 x
2
36. = + ln .
2
2
2 2
2
2
2
x(a + x ) 2a (a + x ) 2a 4 a + x 2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 688