- Iff(x)is periodic with periodL, thenf(x+L) =f(x)for allxand
∫nL
0
f(x)dx=n
∫L
0
f(x)dx,
for integer values ofn.
- ∫x
0
dx
∫x
0
dx···
∫x
︸ ︷︷^0 ︸
n integration signs
dx f(x) =(n−^1 1)!
∫x
0
(x−u)n−^1 f(u)du
Integrals containing algebraic terms
11.
∫ 1
0
xm−^1 (1−x)n−^1 dx=B(m, n) =Γ(Γ(mm)Γ(+nn)), m > 0 , n > 0
12.
∫ 1
0
√dx
1 −x^4
=^1
4
√
2 π
[
Γ(^14 )
] 2
13.
∫ 1
0
dx
(1−x^2 n)n/^2
= 2 nsinπ π
2 n
14.
∫ 1
0
1
β−αx
√ dx
x(1−x)
=√ π
β(β−α)
15.
∫ 1
0
xp−x−p
xq−x−q
dx
x =
π
2 qtan
pπ
2 q, |p|< q
16.
∫ 1
0
xp+x−p
xq+x−q
dx
x =
π
2 qsec
pπ
2 q, |p|< q
17.
∫ 1
0
xp−^1 −x^1 −p
1 −x^2 dx=
π
2 cot
pπ
2 ,^0 < p <^2
18.
∫a
0
√ dx
a^2 −x^2
=π 2
19.
∫a
0
√
a^2 −x^2 dx=π 4 a^2
20.
∫∞
0
dx
x^2 +a^2 =
π
2 a
21.
∫∞
0
xα−^1
1 +xdx=
π
sinαπ,^0 < α <^1
22.
∫ 1
0
xα−^1 +x−α
1 +x dx=
π
sinαπ,^0 < α <^1
23.
∫∞
0
xmdx
1 +x^2 =
π
2 sec
mπ
2
24.
∫∞
0
xα−^1
1 −x^2 dx=
π
2 cot
απ
2
Appendix C
