0-x-2-cosh-x-dx-

Question Number 124251 by bramlexs22 last updated on 02/Dec/20

\underset{0}{\int^{\infty}} \frac{x^{2}}{\cosh x} d x ?

Commented by Dwaipayan Shikari last updated on 02/Dec/20

2 \underset{n = 1}{\sum^{\infty}} {(- 1)}^{n + 1} \int_{0}^{\infty} x^{2} e^{x} e^{- 2 n x} d x \frac{1}{c o s h x} = 2 \frac{e^{x}}{e^{2 x} - 1} = 2 \sum^{\infty} {(- 1)}^{n + 1} e^{x} e^{- 2 n x}

2 \underset{n = 1}{\sum^{\infty}} {(- 1)}^{n + 1} \int_{0}^{\infty} x^{2} e^{- (2 n - 1) x} d x

= 2 \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{{(2 n - 1)}^{3}} \int_{0}^{\infty} u^{2} e^{- u} d u (2 n - 1) x = u

= 2 \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{{(2 n - 1)}^{3}} Γ (3) = 4 \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{{(2 n + 1)}^{3}} = \frac{π^{3}}{8}