Question Number 162876 by HongKing last updated on 01/Jan/22
$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\int}}\:\left(\mathrm{xcot}\boldsymbol{\mathrm{x}}\:\centerdot\:\mathrm{lncos}^{\mathrm{2}} \boldsymbol{\mathrm{x}}\:+\:\mathrm{ln}^{\mathrm{2}} \mathrm{cos}\boldsymbol{\mathrm{x}}\right)\mathrm{dx}\:=\:\frac{\pi^{\mathrm{3}} }{\mathrm{24}} \\ $$