Question Number 164125 by HongKing last updated on 14/Jan/22
$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{arctan}\:\left(\frac{\mathrm{sin}\boldsymbol{\mathrm{x}}}{\mathrm{u}\:+\:\mathrm{cos}\boldsymbol{\mathrm{x}}}\right)\:\mathrm{dudx}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{16}}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{ln}\left(\mathrm{2}\right)\:-\:\frac{\pi}{\mathrm{4}} \\ $$
Answered by Kamel last updated on 14/Jan/22
Commented by HongKing last updated on 15/Jan/22
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Sir}\:\mathrm{cool} \\ $$