Question Number 182271 by SANOGO last updated on 06/Dec/22
Answered by SEKRET last updated on 06/Dec/22
$$\:\:\:\:\frac{−\boldsymbol{\pi}}{\mathrm{2}}\centerdot\boldsymbol{\mathrm{ln}}\left(\mathrm{2}\right) \\ $$
Answered by SEKRET last updated on 06/Dec/22
$$\:\:\:\:\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}}=\int_{\boldsymbol{\mathrm{a}}} ^{\boldsymbol{\mathrm{b}}} \boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}−\boldsymbol{\mathrm{x}}\right)\:\boldsymbol{\mathrm{dx}} \\ $$$$\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sinx}}\right)\:\boldsymbol{\mathrm{dx}}\:=\:\:\boldsymbol{\mathrm{I}} \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right)\:\boldsymbol{\mathrm{dx}}=\:\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\frac{\boldsymbol{\mathrm{sin}}\left(\mathrm{2}\boldsymbol{\mathrm{x}}\right)}{\mathrm{2}}\right)\:\boldsymbol{\mathrm{dx}}\:=\:\:\mathrm{2}\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sin}}\left(\mathrm{2}\boldsymbol{\mathrm{x}}\right)\right)\:\boldsymbol{\mathrm{dx}}\:−\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\mathrm{2}\right)\boldsymbol{\mathrm{dx}}=\mathrm{2}\boldsymbol{\mathrm{I}} \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\boldsymbol{\pi}} \frac{\mathrm{1}}{\mathrm{2}}\centerdot\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)\right)\:\boldsymbol{\mathrm{dx}}\:−\frac{\boldsymbol{\pi}}{\mathrm{2}}\centerdot\boldsymbol{\mathrm{ln}}\left(\mathrm{2}\right)=\:\mathrm{2}\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\centerdot\left(\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sinx}}\right)\boldsymbol{\mathrm{dx}}+\int_{\frac{\boldsymbol{\pi}}{\mathrm{2}}} ^{\:\boldsymbol{\pi}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sinx}}\right)\boldsymbol{\mathrm{dx}}\right)−\frac{\boldsymbol{\pi}\centerdot\boldsymbol{\mathrm{ln}}\left(\mathrm{2}\right)}{\mathrm{2}}=\mathrm{2}\boldsymbol{\mathrm{I}} \\ $$$$\:\:\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{\mathrm{I}}+\boldsymbol{\mathrm{I}}\right)\:−\:\frac{\boldsymbol{\pi}}{\mathrm{2}}\centerdot\boldsymbol{\mathrm{ln}}\left(\mathrm{2}\right)=\mathrm{2}\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{I}}\:=\:\frac{−\pi}{\mathrm{2}}\:\centerdot\boldsymbol{\mathrm{ln}}\left(\mathrm{2}\right) \\ $$$$\:\:\boldsymbol{\mathrm{ABDULAZIZ}}\:\:\:\:\boldsymbol{\mathrm{ABDUVALIYEV}} \\ $$
Commented by SEKRET last updated on 06/Dec/22
$$\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sinx}}\right)\boldsymbol{\mathrm{dx}}=\int_{\frac{\boldsymbol{\pi}}{\mathrm{2}}} ^{\:^{\boldsymbol{\pi}} } \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{sinx}}\right)\boldsymbol{\mathrm{dx}}=\boldsymbol{\mathrm{I}} \\ $$
Commented by SANOGO last updated on 06/Dec/22
$${thank}\:{you} \\ $$