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

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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

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

$\blacksquare$


Sources