# Definite Integral to Infinity of x over Hyperbolic Sine of a x

## Theorem

$\displaystyle \int_0^\infty \frac x {\sinh a x} \rd x = \frac {\pi^2} {4 a^2}$

where $a$ is a positive real number.

## Proof

 $\ds \int_0^\infty \frac x {\sinh a x} \rd x$ $=$ $\ds \frac {2^2 - 1} {2 a^2} \map \Gamma 2 \map \zeta 2$ Definite Integral to Infinity of $\dfrac {x^n} {\sinh a x}$ $\ds$ $=$ $\ds \frac 3 {2 a^2} \times 1! \times \frac {\pi^2} 6$ Gamma Function Extends Factorial, Basel Problem $\ds$ $=$ $\ds \frac \pi {4 a^2}$

$\blacksquare$