Table of Integrals, Series, and Products, 8th Edition

4

Definite Integrals of Elementary Functions

4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers

4.111

1. 

$\begin{array}{lll}{\displaystyle {\int }_0^{\infty }\frac{sin\text{?}ax}{sinh\text{?}bx}{x}^{2m}\text{?}\text{d}x={\left(-1\right)}^m\frac{\pi }{2b}\frac{{\partial }^{2m}}{\partial a^{2m}}\left(tanh\text{?}\frac{a\pi }{2b}\right)}\hfill & \left[\mathrm{Re}b>0\right]\hfill & \left(\text{cf}\text{.}\text{?}\text{3}\text{.981}\text{?}\text{1}\right)\hfill \end{array}$

  GW (336)(17a)

2. 

$\begin{array}{lll}{\displaystyle {\int }_0^{\infty }\frac{cos\text{?}ax}{sinh\text{?}bx}{x}^{2m+1}\text{?}\text{d}x={\left(-1\right)}^m\frac{\pi }{2b}\frac{{\partial }^{2m+1}}{\partial a^{2m+1}}\left(tanh\text{?}\frac{a\pi }{2b}\right)}\hfill & \left[\mathrm{Re}b>0\right]\hfill & \left(\text{cf}\text{.}\text{?}\text{3}\text{.981}\text{?}\text{1}\right)\hfill \end{array}$

  GW (336)(17b)

3. 

$\begin{array}{lll}{\displaystyle {\int }_0^{\infty }\frac{sin\text{?}ax}{cosh\text{?}bx}{x}^{2m+1}\text{?}\text{d}x={\left(-1\right)}^{m+1}\frac{\pi }{2b}\frac{{\partial }^{2m+1}}{\partial a^{2m+1}}\left(\frac{1}{cosh\text{?}\frac{a\pi }{2b}}\right)}\hfill & \left[\mathrm{Re}b>0\right]\hfill & \left(\text{cf}\text{.}\text{?}\text{3}\text{.981}\text{?}3\right)\hfill \end{array}$

  GW (336)(18b)

4. 

$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\frac{cos\text{?}ax}{cosh\text{?}bx}{x}^{2m}\text{?}\text{d}}\hfill \end{array}$