Definite Integral to Infinity of x over Hyperbolic Sine of a x
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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: $15.114$