Page 705 - Handbook Of Integral Equations

dx 1 1 1 x
2
37. = + + ln .
2 3
2
2
2
2
2 2
2
2
4
x(a + x ) 4a (a + x ) 2a (a + x ) 2a 6 a + x 2
dx 1 1 x

38. = – – arctan .
2
2
2
2
x (a + x ) a x a 3 a
dx 1 x 3 x

39. = – – – arctan .
4
2 2
2
x (a + x ) a x 2a (a + x ) 2a 5 a
2
2
2
4
dx 1 1 1 x
2
40. = – – – ln .
2
4 2
2
4
2 2
3
2
2
x (a + x ) 2a x 2a (a + x ) a 6 a + x 2
dx 1 x 7x 15 x

41. = – – – – arctan .
6
2
2
2
2
4
2
2 3
6
2 2
x (a + x ) a x 4a (a + x ) 8a (a + x ) 8a 7 a
dx 1 1 1 3 x
2
42. = – – – – ln .
2
6
6 2
2 3
3
2
4
2
2
2
2 2
x (a + x ) 2a x a (a + x ) 4a (a + x ) 2a 8 a + x 2
2
2
Integrals containing a – x .
dx 1

43. = a + x .
2
a – x 2 2a ln a – x
dx x 1

44. = + a + x .
2
2 2
2
2
2
(a – x ) 2a (a – x ) 4a 3 ln a – x

dx x 3x 3
45. = + + ln a + x .
(a – x ) 4a (a – x ) 8a (a – x ) 16a 5 a – x
2
2 2
2 3
4
2
2
2
2
dx x 2n – 1 dx

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

47. = – ln |a – x |.
2
a – x 2 2
xdx 1

48. = .
2
2
2
2 2
(a – x ) 2(a – x )
xdx 1

49. = .
2 2
2
2 3
2
(a – x ) 4(a – x )
xdx 1

50. = 2 n ; n =1, 2, ...
2 n+1
2
(a – x ) 2n(a – x )
2
x dx a
2
51. = –x + a + x .
2
a – x 2 2 ln a – x
2
x dx x 1
52. = – ln a + x .
2
2
2
2 2
(a – x ) 2(a – x ) 4a a – x
x dx x x 1
2
53. = – – a + x .
2
2
2 2
2
2
2 3
2
(a – x ) 4(a – x ) 8a (a – x ) 16a 3 ln a – x
x dx x 1 dx
2
54. = – ; n =1, 2, ...
2 n
2
2 n+1
2 n
2
2
(a – x ) 2n(a – x ) 2n (a – x )
3 2 2
x dx x a 2 2
55. = – – ln |a – x |.
2
a – x 2 2 2
3 2
x dx a 1 2 2
56. = + ln |a – x |.
2
2
2 2
2
(a – x ) 2(a – x ) 2
3 2
x dx 1 a
57. = – + ; n =2, 3, ...
2 n
2
2
(a – x ) 2(n – 1)(a – x ) 2n(a – x )
2 n+1
2
2 n–1
dx 1 x
2
58. = .

2
2
2
x(a – x ) 2a 2 ln a – x 2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 689