Page 707 - Handbook Of Integral Equations

4
4
Integrals containing a ± x .
√ √
dx 1 a + ax 2+ x 1 ax 2
2 2
79. = √ ln √ + √ arctan .
4
2
a + x 4 4a 3 2 a – ax 2+ x 2 2a 3 2 a – x 2
2
xdx 1 x
2
80. = arctan .
a + x 4 2a 2 a 2
4
√ √
x dx 1 a + ax 2+ x 1 ax 2
2 2 2
81. = – √ ln √ + √ arctan .
2
4
a + x 4 4a 2 a – ax 2+ x 2 2a 2 a – x 2
2

dx 1 1 x
82. = ln a + x + arctan .
4
a – x 4 4a 3 a – x 2a 3 a
xdx 1 2
2
83. = a + x .
2
4
a – x 4 4a 2 ln a – x 2
x dx 1 1 x
2
84. = a + x – arctan .
4
a – x 4 4a ln a – x 2a a
dx 1 x
m
85. = ln .

m
x(a + bx ) am a + bx m
2.2. Integrals Containing Irrational Functions
Integrals containing x 1/2 .
x dx 2 1/2 2a bx
1/2 1/2
1. = x – arctan .
2
2
a + b x b 2 b 3 a
x dx 2x 2a x 2a bx
3/2 3/2 2 1/2 3 1/2
2. = – + arctan .
2
a + b x 3b 2 b 4 b 5 a
2
x dx x 1 bx
1/2 1/2 1/2
3. = – + arctan .
2
2
2
2
2
(a + b x) 2 b (a + b x) ab 3 a
x dx 2x 3a x 3a bx
3/2 3/2 2 1/2 1/2
4. = + – arctan .
2
2
2
2
2
4
2
2
(a + b x) 2 b (a + b x) b (a + b x) b 5 a
dx 2 bx
1/2
5. = arctan .
2
2
(a + b x)x 1/2 ab a
dx 2 2b bx
1/2
6. = – – arctan .
2 1/2
2
2
(a + b x)x 3/2 a x a 3 a
1/2 1/2
dx x 1 bx
7. = + arctan .
2
2
3
2
2
2
2 1/2
(a + b x) x a (a + b x) a b a
1/2 1/2
x dx 2 1/2 2a a + bx
8. = – x + .

2
2
a – b x b 2 b 3 ln a – bx 1/2
x dx 2x 2a x a a + bx
3/2 3/2 2 1/2 3 1/2
9. = – – + ln .

2
2
a – b x 3b 2 b 4 b 5 a – bx 1/2
x dx x 1 a + bx 1/2
1/2 1/2

10. = – .

2
2
2
2
2
(a – b x) 2 b (a – b x) 2ab 3 ln a – bx 1/2
3/2 2 1/2 2 3/2 1/2
x dx 3a x – 2b x 3a a + bx
11. = – ln .

2
2
(a – b x) 2 b (a – b x) 2b 5 a – bx 1/2
2
2
4

dx 1 a + bx 1/2
12. = ln .

2
2
(a – b x)x 1/2 ab a – bx 1/2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 691