Page 721 - Handbook Of Integral Equations

∞
x dx m+1 (2n – 2m – 3)!! a
m m–n+1/2
34. n+1/2 =2 m! m+1 , a, b >0,
0 (a + bx) (2n – 1)!! b
1
n, m =1, 2, ... , m < b – .
2
∞

dx π (2n – 3)!! 1
35. 2 2 n = 2n–1 , n =1, 2, ...
0 (x + a ) 2 (2n – 2)!! a
∞
(x +1) 1 – a
λ–1 –λ
36. λ+1 dx = , a >0.
0 (x + a) λ(a – 1)
1
x λ–1 dx π
37. λ = λ , 0 < λ <1, a > –1.
0 (1 + ax)(1 – x) (1 + a) sin(πλ)
1 x λ–1/2 dx sin[(2λ – 1)k] √

Γ 1 – λ cos
38. λ λ =2π –1/2 Γ λ + 1 2 2λ k , k = arctan a;
0 (1 + ax) (1 – x) (2λ – 1) sin k
– 1 < λ <1, a >0.
2
∞ x dx π
a–1
39. b = , 0 < a ≤ b.
0 x +1 b sin(πa/b)
∞ x dx π(a – b)
a–1
40. b 2 = 2 , a <2b.
0 (x +1) b sin[π(a – b)/b]
∞ x dx √ √ √ 1–2λ Γ(λ – 1/2)
λ–1/2
41. λ λ = π a + b , λ >0.
0 (x + a) (x + b) Γ(λ)
∞ 1 – x π sin A πa πc
a
42. b x c–1 dx = , A = , C = ; a + c < b, c >0.
0 1 – x b sin C sin(A + C) b b
a–1
∞ x dx
1
1
43. 2 1–b = B 1 2 a,1 – b – a , 1 2 a + b <1, a >0.
2
2
0 (1 + x )
2m
∞ x dx π(2m – 1)!! (2n – 2m – 3)!!
44. 2 n = √ , a, b >0, n > m +1.
m n–m–1
0 (ax + b) 2(2n – 2)!! a b ab
∞
2m+1
x dx m!(n – m – 2)!
45. 2 n = m+1 n–m–1 , ab >0, n > m +1 ≥ 1.
0 (ax + b) 2(n – 1)!a b
∞ x dx 1 µ µ
µ–1
46. = B , ν – , p >0, 0 < µ < pν.
p ν
(1 + ax ) pa µ/p p p
0
n+1
∞ √ na
n
47. x + a – x dx = , n =2, 3, ...
2
2
2
n – 1
0
∞ dx n

48. √ = n–1 2 , n =2, 3, ...
n
2
0 x + x + a 2 a (n – 1)
∞ √ n n ⋅ m! a
n+m+1
2
2
49. x m x + a – x dx = ,
0 (n – m – 1)(n – m +1) ... (n + m +1)
n, m =1, 2, ... , 0 ≤ m ≤ n – 2
∞ x dx n ⋅ m!
m
50. √ = n–m–1 , n =2, 3, ...
n
2
0 x + x + a 2 (n – m – 1)(n – m +1) ... (n + m +1)a
3.2. Integrals Containing Exponential Functions
∞ 1

1. e –ax dx = , a >0.
0 a
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 705