Page 722 - Handbook Of Integral Equations

1 n
n! n! 1
n –ax
2. x e dx = – e –a , a >0, n =1, 2, ...
a n+1 k! a n–k+1
0 k=0
∞ n!

n –ax
3. x e dx = , a >0, n =1, 2, ...
a n+1
0
∞
e π
–ax
4. √ dx = , a >0.
0 x a
∞ Γ(ν)

e
5. x ν–1 –µx dx = ν , µ, ν >0.
0 µ
∞ dx ln 2

6. ax = .
0 1+ e a
∞ 2n
x dx n–1 2π B 2n
2n–1
7. px =(–1) , n =1, 2, ... (B m are the Bernoulli numbers).
0 e – 1 p 4n
∞ 2n |B 2n |
x dx 1–2n 2π
2n–1
8. =(1 – 2 ) , n =1, 2, ...
e px +1 p 4n
0
∞ e dx π
–px
9. = , q > p >0 or 0> p > q.
1+ e –qx q sin(πp/q)
–∞
∞
ax –ax
e + e π
10. bx –bx dx = πa , b > a.
0 e + e 2b cos
2b
∞
e – e π πp
–px –qx
11. –(p+q)x dx = cot , p, q >0.
0 1 – e p + q p + q
∞

–βx ν –µx 1 µ

12. 1 – e e dx = B , ν +1 .
0 β β

∞ 1 π

2
13. exp –ax dx = , a >0.
0 2 a
∞ n!

14. x 2n+1 exp –ax 2 dx = n+1 , a >0, n =1, 2, ...
0 2a
√
∞ 1 ⋅ 3 ... (2n – 1) π

15. x 2n exp –ax 2 dx = , a >0, n =1, 2, ...
a
2 n+1 n+1/2
0
√
∞ π b 2

2 2
16. exp –a x ± bx dx = |a| exp 2 .
–∞ 4a
∞ b 1 π √

2
17. exp –ax – dx = exp –2 ab , a, b >0.
x 2 2 a
0
∞

a 1 1
18. exp –x dx = Γ , a >0.
a a
0
3.3. Integrals Containing Hyperbolic Functions
∞
dx π

1. = 2|a| .
0 cosh ax
√
2 b – a
 2 2

 √ arctan if |b| > |a|,
∞ dx  2 2 a + b

2. = b – a √
2
0 a + b cosh x  1 a + b + a – b 2
 ln if |b| < |a|.
 √ √
2
2
a – b 2 a + b – a + b 2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 706