Page 720 - Handbook Of Integral Equations

a
1 x dx πa
14. a = , –1< a <1.
0 (1 – x) sin(πa)
1 Γ(p)Γ(q)
15. x p–1 (1 – x) q–1 dx ≡ B(p, q)= , p, q >0.
0 Γ(p + q)
1
π
q –p/q
16. x p–1 (1 – x ) dx = , q > p >0.
0 q sin(πp/q)
1
πp
q –p/q
17. x p+q–1 (1 – x ) dx = 2 , q > p.
0 q sin(πp/q)
1 π
q –1/p
18. x q/p–1 (1 – x ) dx = , p >1, q >0.
0 q sin(π/p)
1 x p–1 – x –p
19. dx = π cot(πp), |p| <1.
0 1 – x
1 x p–1 – x –p π
20. dx = , |p| <1.
0 1+ x sin(πp)
1
p
x – x –p 1
21. dx = – π cot(πp), |p| <1.
x – 1 p
0
p
1 x – x –p 1 π
22. dx = – , |p| <1.
0 1+ x p sin(πp)
1 x 1+p – x 1–p π πp 1
23. 2 dx = cot – , |p| <1.
0 1 – x 2 2 p
x – x 1 π
1 1+p 1–p
24. dx = – , |p| <1.
1+ x 2 p 2 sin(πp/2)
0
∞ x – x
p–1 q–1
25. dx = π[cot(πp) – cot(πq)], p, q >0.
0 1 – x
1
dx 2
26. = arctan a.
2
0 (1 + a x)(1 – x) a
1
dx 1 1+ a
27. = ln .
2
0 (1 – a x)(1 – x) a 1 – a
1 dx π
28. √ = √ , 1 < a.
2
–1 (a – x) 1 – x 2 a – 1
n
1 x dx 2(2n)!!
29. √ = , n =1, 2, ...
0 1 – x (2n + 1)!!
1 x n–1/2 dx π (2n – 1)!!
30. √ = , n =1, 2, ...
0 1 – x (2n)!!
x dx π 1 ⋅ 3 ... (2n – 1)
1 2n
31. √ = , n =1, 2, ...
0 1 – x 2 2 2 ⋅ 4 ... (2n)
1
x 2n+1 dx 2 ⋅ 4 ... (2n)
32. √ = , n =1, 2, ...
0 1 – x 2 1 ⋅ 3 ... (2n +1)
∞
λ–1 n
x dx n πC λ–1
33. n+1 =(–1) λ , 0 < λ < n +1.
0 (1 + ax) a sin(πλ)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 704