Definite Integral to Infinity of x over Exponential of x plus One
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Theorem
- $\ds \int_0^\infty \frac x {e^x + 1} \rd x = \frac {\pi^2} {12}$
Proof
\(\ds \int_0^\infty \frac x {e^x + 1} \rd x\) | \(=\) | \(\ds \int_0^\infty \frac {x^{2 - 1} } {e^x + 1} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \eta 2 \map \Gamma 2\) | Integral Representation of Dirichlet Eta Function in terms of Gamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^2} } 1!\) | Definition of Dirichlet Eta Function, Gamma Function Extends Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\pi^2} {12}\) | Sum of Reciprocals of Squares Alternating in Sign |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.81$