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

Theorem

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

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

\(\ds \)

\(=\)

\(\ds \frac \pi {4 a^2}\)

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Hyperbolic Functions: $15.114$