prove-that-0-x-2-cosh-x-dx-pi-3-8-

Question Number 174685 by mnjuly1970 last updated on 08/Aug/22

p r o v e t h a t :

Ω = \int_{0}^{\infty} \frac{x^{2}}{c o s h (x)} d x = \frac{π^{3}}{8}

Answered by princeDera last updated on 08/Aug/22

\frac{1}{co s h (x)} = \frac{2 e^{- x}}{1 + e^{- 2 x}}

Ω = \int_{0}^{\infty} \frac{2 x^{2} e^{- x}}{1 + \cosh (x)} d x = 2 \int_{0}^{\infty} x^{2} e^{- x} \sum_{k ⩾ 0} {(- 1)}^{k} e^{- 2 k x} d x

= 2 \sum_{k ⩾ 0} {(- 1)}^{k} \int_{0}^{\infty} x^{2} e^{- (2 k + 1) x} d x = 4 \sum_{k ⩾ 0} \frac{{(- 1)}^{k}}{{(2 k + 1)}^{3}}

4 B (3) = 4 (\frac{π^{3}}{32}) = \frac{π^{3}}{8}

w h e r e B (z) i s t h e d i r i c h l e t b e t a f u n c t i o n

Commented by MME last updated on 08/Aug/22

B o s s m i

Commented by mnjuly1970 last updated on 08/Aug/22

t h a n k s a l o t s i r

j u s t, β (3) = \frac{π^{3}}{32}

t y p o \dots

Commented by princeDera last updated on 08/Aug/22

c o r r e c t e d .

t h a n k y o u

Commented by princeDera last updated on 08/Aug/22

m y b o s s