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Question Number 160636 by mnjuly1970 last updated on 03/Dec/21

$$$$$$provethat$$$$$$$${\int }_0^1{\left(\frac{{tanh}^{-1}\left(x\right)}{x}\right)}^2=\zeta \left(2\right)◼m.n$$$$$$$$$$

Answered by Kamel last updated on 03/Dec/21

$$\stackrel{t=\frac{1-x}{1+x}}{=}\frac{1}{2}{\int }_0^1\frac{{Ln}^2\left(t\right)}{{(1-t)}^2}dt\stackrel{IBP}{=}-{\int }_0^1\frac{Ln\left(t\right)}{(1-t)}dt=\zeta \left(2\right)$$$$$$

Commented by mnjuly1970 last updated on 04/Dec/21

$$merceysirkamel$$

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