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mathematics-prove-that-0-arctan-x-ln-1-x-2-1-x-2-dx-pi-2-ln-2-4-7-8-3-




Question Number 138940 by mnjuly1970 last updated on 20/Apr/21
                     ... mathematics..  prove that:  𝛗=∫_0 ^( ∞) ((arctan(x).ln(1+x^2 ))/(1+x^2 ))dx     =((Ο€^2 ln(2))/4)+(7/8) ΞΆ(3)
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{mathematics}.. \\ $$$${prove}\:{that}: \\ $$$$\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{arctan}\left({x}\right).{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$$\:\:\:=\frac{\pi^{\mathrm{2}} {ln}\left(\mathrm{2}\right)}{\mathrm{4}}+\frac{\mathrm{7}}{\mathrm{8}}\:\zeta\left(\mathrm{3}\right) \\ $$

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