Question Number 170384 by amin96 last updated on 22/May/22
$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}} \\ $$$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{x}}\right)\right)\boldsymbol{\mathrm{dx}}=\pi\boldsymbol{\mathrm{ln}}\left(\sqrt{\frac{\boldsymbol{\mathrm{a}}−\mathrm{1}}{\boldsymbol{\mathrm{a}}}}+\mathrm{1}\right)+\frac{\pi}{\mathrm{2}}\boldsymbol{\mathrm{ln}}\left(\frac{\boldsymbol{\mathrm{a}}}{\mathrm{4}}\right) \\ $$$$\boldsymbol{\mathrm{a}}>\mathrm{0}\:\:\:\:\boldsymbol{\mathrm{I}{m}}\left\{\boldsymbol{{a}}\right\}\neq\mathrm{0}\:\: \\ $$