Page 709 - Handbook Of Integral Equations

2 1/2
2
Integrals containing (a – x ) .
2
1 2 2 1/2 a x
2 1/2
2
32. (a – x ) dx = x(a – x ) + arcsin .
2 2 a
1 2 2 3/2

2 1/2
2
33. x(a – x ) dx = – (a – x ) .
3

1 3 3 x
2 1/2
2 3/2
4
2
2
2
2 3/2
2
34. (a – x ) dx = x(a – x ) + a x(a – x ) + a arcsin .
4 8 8 a
2 2 1/2
1 2 2 1/2 2 2 1/2 a +(a – x )
35. (a – x ) dx =(a – x ) – a ln .

x x
dx x

36. √ = arcsin .
2
a – x 2 a
xdx 2 2 1/2

37. √ = –(a – x ) .
2
a – x 2

2
2 –1/2
–2
2
2 –3/2
38. (a – x ) dx = a x(a – x ) .
Reduction formulas. The parameters a, b, p, m, and n below can assume arbitrary values,
except for those at which denominators vanish in successive applications of a formula. Notation:
n
w = ax + b.
1
n
p
m
m
p
39. x (ax + b) dx = x m+1 w + npb x w p–1 dx .
m + np +1
1 m+1 p+1 m p+1

m
n
p
40. x (ax + b) dx = –x w +(m + n + np +1) x w dx .
bn(p +1)
1
n
m
p
p
41. x (ax + b) dx = x m+1 w p+1 – a(m + n + np +1) x m+n w dx .
b(m +1)
1 m–n+1 p+1 m–n p

p
n
m
42. x (ax + b) dx = x w –b(m – n +1) x w dx .
a(m + np +1)
2.3. Integrals Containing Exponential Functions
1 ax

ax
1. e dx = e .
a
a x
x
2. a dx = .
ln a

x 1
3. xe ax dx = e ax – .
a a 2
2
x 2x 2
2 ax
4. x e dx = e ax – + .
a a 2 a 3

1 n n(n – 1) n! n!
n
n ax
5. x e dx=e ax x – x n–1 + x n–2 –···+(–1) n–1 x+(–1) n , n=1, 2, ...
a a 2 a 3 a n a n+1
n
(–1) k d k
6. P n (x)e ax dx=e ax P n (x), where P n (x) is an arbitrary nth-degree polynomial.
a k+1 dx k
k=0

dx x 1 px
7. = – ln |a + be |.
a + be px a ap
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 693