DEFINITE INTEGRALS CONTAINING HYPERBOLIC FUNCTIONS

1.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin ax}{\sinh bx}dx=\displaystyle \frac{\pi}{2b}\tanh\displaystyle \frac{a\pi}{2b}$

2.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\cos ax}{\cosh bx}dx=\displaystyle \frac{\pi}{2b}\displaystyle \frac{1}{\cosh (a\pi/2b)}$

3.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{x dx}{\sinh ax}=\displaystyle \frac{\pi^2}{4a^2}$

4.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{x^n dx}{\sinh ax}=\displaysty... ...tyle \frac{1}{2^{n+1}}+\displaystyle \frac{1}{3^{n+1}}+\cdot\cdot\cdot \right\}$

5.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sinh ax}{e^{bx}+1}dx=\displaystyle \frac{\pi}{2b}\csc\displaystyle \frac{a\pi}{b}-\displaystyle \frac{1}{2a}$

6.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sinh ax}{e^{bx}-1}dx=\displaystyle \frac{1}{2a}-\displaystyle \frac{\pi}{2b}\cot\displaystyle \frac{a\pi}{b}$