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

Theorem

 * $\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$

where:
 * $a$ and $n$ are positive real numbers
 * $\Gamma$ denotes the gamma function
 * $\zeta$ denotes the Riemann zeta function.

Proof
We have:

So:

giving:


 * $\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$