0-x-2-ln-1-e-x-x-dx-7pi-4-360-

Question Number 140221 by qaz last updated on 05/May/21

\int_{0}^{\infty} x^{2} [l n (1 + e^{x}) - x] d x = \frac{7 π^{4}}{360}

Answered by Dwaipayan Shikari last updated on 05/May/21

\int_{0}^{\infty} x^{2} l o g (\frac{1 + e^{x}}{e^{x}}) d x

= Σ \int_{0}^{\infty} {(- 1)}^{n} x^{2} \frac{e^{- n x}}{n} d x

= Σ \frac{{(- 1)}^{n + 1}}{n^{4}} Γ (3) = 2 ζ (4) (1 - \frac{1}{2^{3}}) = \frac{7}{4} ζ (4) = \frac{7 π^{4}}{360}

Commented by qaz last updated on 05/May/21

t h a n k y o u s i r

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