Page 723 - Handbook Of Integral Equations

∞ x dx π 2n+1
2n
3. = |E 2n |, a >0.
0 cosh ax 2a
∞ x π (2 – 2)
2n 2n 2n
4. 2 dx = 2n |B 2n |, a >0.
0 cosh ax a(2a)
∞ cosh ax π

5. dx = πa , b > |a|.
0 cosh bx 2b cos
2b
2n
∞ cosh ax π d 1
6. x 2n dx = 2n 1 , b > |a|, n =1, 2, ...
0 cosh bx 2b da cos 2 πa/b
πa πb

∞
cos cos
cosh ax cosh bx π 2c 2c
7. dx = , c > |a| + |b|.
0 cosh(cx) c cos πa + cos πb
c c
∞ xdx π
2
8. = , a >0.
sinh ax 2a 2
0
√
∞ dx 1 a + b + a + b 2
2

9. = √ ln √ , ab ≠ 0.
2
2
0 a + b sinh x a + b 2 a + b – a + b 2

∞ sinh ax π πa
10. dx = tan , b > |a|.
0 sinh bx 2b 2b
∞ sinh ax π d πa
2n
11. x 2n dx = 2n tan , b > |a|, n =1, 2, ...
0 sinh bx 2b dx 2b
∞ x π
2n 2n
12. 2 dx = 2n+1 |B 2n |, a >0.
0 sinh ax a
3.4. Integrals Containing Logarithmic Functions
1

n
n
1. x a–1 ln xdx =(–1) n! a –n–1 , a >0, n =1, 2, ...
0
ln x π
1 2
2. dx = – .
0 x +1 12
x ln x n+1 π (–1)
1 n 2 n k
3. dx =(–1) + 2 , n =1, 2, ...
0 x +1 12 k
k=1
1 x µ–1 ln x πa µ–1

4. dx = ln a – π cot(πµ) , 0 < µ <1.
0 x + a sin(πµ)
1
µ
5. |ln x| dx = Γ(µ + 1), µ > –1.
0
∞ π

6. x µ–1 ln(1 + ax) dx = µ , –1< µ <0.
0 µa sin(πµ)
1 2n
1 (–1) k–1
7. x 2n–1 ln(1 + x) dx = , n =1, 2, ...
0 2n k
k=1
1 1 2n+1 (–1) k

8. x 2n ln(1 + x) dx = ln 4 + , n =0, 1, ...
0 2n +1 k
k=1
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 707