Question Number 160636 by mnjuly1970 last updated on 03/Dec/21
$$ \\ $$$$\:\:\:\:\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}\:\:} ^{\:\mathrm{1}} \left(\frac{{tanh}^{\:−\mathrm{1}} \left(\:{x}\:\right)}{{x}}\:\right)^{\:\mathrm{2}} =\:\zeta\:\left(\:\mathrm{2}\:\right)\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\: \\ $$$$ \\ $$$$ \\ $$
Answered by Kamel last updated on 03/Dec/21
$$\overset{{t}=\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}} {=}\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{Ln}^{\mathrm{2}} \left({t}\right)}{\left(\mathrm{1}−{t}\right)^{\mathrm{2}} }{dt}\overset{{IBP}} {=}−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{Ln}\left({t}\right)}{\left(\mathrm{1}−{t}\right)}{dt}=\zeta\left(\mathrm{2}\right) \\ $$$$ \\ $$
Commented by mnjuly1970 last updated on 04/Dec/21
$${mercey}\:{sir}\:{kamel} \\ $$