Page 703 - Handbook Of Integral Equations

Supplement 2

Tables of Indefinite Integrals

2.1. Integrals Containing Rational Functions

Integrals containing a + bx.*
dx 1

1. = ln |a + bx|.
a + bx b
n+1
(a + bx)
n
2. (a + bx) dx = , n ≠ –1.
b(n +1)
xdx 1

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

4. = (a + bx) – 2a(a + bx)+ a ln |a + bx| .
a + bx b 3 2
dx 1

5. = – ln a + bx .
x(a + bx) a x
dx 1 b

6. = – + a + bx .
2
x (a + bx) ax a 2 ln x

xdx 1 a
7. = ln |a + bx| + .
(a + bx) 2 b 2 a + bx
x dx 1 a
2 2
8. = a + bx – 2a ln |a + bx| – .
(a + bx) 2 b 3 a + bx

dx 1 1
9. = – a + bx .
x(a + bx) 2 a(a + bx) a 2 ln x

xdx 1 1 a
10. = – + .
(a + bx) 3 b 2 a + bx 2(a + bx) 2
Integrals containing a + x and b + x.

a + x
11. dx = x +(a – b)ln |b + x|.
b + x
dx 1

12. = b + x , a ≠ b.For a = b, see integral 2 with n = –2.
(a + x)(b + x) a – b ln a + x
xdx 1

13. = a ln |a + x| – b ln |b + x| .
(a + x)(b + x) a – b
dx 1 1

14. = + ln a + x .
(a + x)(b + x) 2 (b – a)(b + x) (a – b) 2 b + x
* Throughout this section, the integration constant C is omitted for brevity.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 687