Page 725 - Handbook Of Integral Equations

∞ Γ(1/p) π

p
14. cos ax dx = 1/p cos , a >0, p >1.
0 pa 2p
π/2
π 1 ⋅ 3 ... (2n – 1)
15. sin 2n xdx = , n =1, 2, ...
2 2 ⋅ 4 ... (2n)
0
π/2
2 ⋅ 4 ... (2n)
16. sin 2n+1 xdx = , n =1, 2, ...
1 ⋅ 3 ... (2n +1)
0
∞ sin ax π

17. dx = sign a.
x 2
0
∞ sin ax π
2
18. 2 dx = |a|.
0 x 2
∞ sin ax π

19. √ dx = , a >0.
x 2a
0
π
π 2 Γ(µ +1)
µ
20. x sin xdx = µ+1 , µ > –1.
2
Γ µ +
0 2 1
2
∞

–µ
21. x µ–1 sin ax dx = a Γ(µ) sin 1 πµ , a >0, 0 < µ <1.
2
0
π/2
sin xdx 1 1+ k
22. √ = ln .
2
2
0 1 – k sin x 2k 1 – k
∞

2 1 π
23. sin ax dx = , a >0.
2 2a
0
∞

p Γ(1/p) π
24. sin ax dx = sin , a >0, p >1.
pa 1/p 2p
0
π/2 n! m!
25. sin 2n+1 x cos 2m+1 xdx = , n, m =1, 2, ...
0 2(n + m + 1)!
π/2
1
1

26. sin p–1 x cos q–1 xdx = B 1 p, q .
2 2 2
0
2π

2

27. (a sin x + b cos x) 2n dx =2π (2n – 1)!! a + b 2 n , n =1, 2, ...
0 (2n)!!
 π
 2 if |a| <1,

∞ sin x cos ax
28. dx = π 4 if |a| =1,
0 x 
0 if 1 < |a|.
π sin xdx 2 if 0 ≤ a ≤ 1,
29. √ =
2
0 a +1 – 2a cos x 2/a if 1 < a.
∞ tan ax π

30. dx = sign a.
0 x 2
π/2 π
31. (tan x) ±λ dx = 1 , |λ| <1.
0 2 cos 2 πλ
∞ b

32. e –ax sin bx dx = 2 2 , a >0.
0 a + b
∞ a

33. e –ax cos bx dx = 2 2 , a >0.
0 a + b
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 709