Page 719 - Handbook Of Integral Equations

Supplement 3

Tables of Definite Integrals

Throughout Supplement 3 it is assumed that n is a positive integer, unless otherwise specified.

3.1. Integrals Containing Power-Law Functions

∞ dx π
1. 2 = √ .
0 ax + b 2 ab
√
∞
dx π 2

2. 4 = .
0 x +1 4
x dx n (–1)
1 n n k
3. =(–1) ln 2 + .
0 x +1 k
k=1
∞ x dx π
a–1
4. = , 0 < a <1.
0 x +1 sin(πa)
∞ x dx π(1 – λ)
λ–1
5. 2 = λ , 0 < λ <2.
0 (1 + ax) a sin(πλ)
1 dx β
6. 2 = .
0 x +2x cos β +1 2 sin β
1 a –a
x + x dx π sin(aβ)
7. = , |a| <1, β ≠ (2n +1)π.
2
x +2x cos β +1 sin(πa) sin β
0
∞ x dx π(a – b )
λ–1 λ–1 λ–1
8. = , 0 < λ <2.
0 (x + a)(x + b) (b – a) sin(πλ)
λ–1
∞ x (x + c) dx π a – c b – c
9. = a λ–1 + b λ–1 , 0 < λ <1.
0 (x + a)(x + b) sin(πλ) a – b b – a
∞ x dx πλ(1 – λ)
λ
10. 3 = , –1< λ <2.
0 (x +1) 2 sin(πλ)
∞ x dx π b – a
λ–1 λ–2 λ–2
11. 2 2 2 2 = , 0 < λ <4.
2
0 (x + a )(x + b ) 2 a – b 2 sin(πλ/2)
1
πa(1 – a)
a
12. x (1 – x) 1–a dx = , –1< a <1.
0 2 sin(πa)
dx π
1
13. a 1–a = , 0 < a <1.
0 x (1 – x) sin(πa)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 703