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4

# Definite Integrals of Elementary Functions

## 4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers

4.111

1.

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

GW (336)(17a)

2.

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

GW (336)(17b)

3.

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

GW (336)(18b)

4.

$\begin{array}{l}{\int }_{0}^{\infty }\frac{cos\text{?}ax}{cosh\text{?}bx}{x}^{2m}\text{?}\text{d}\hfill \end{array}$

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