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

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