Question Number 24852 by nnnavendu last updated on 27/Nov/17
$$\mathrm{please}\:\mathrm{prove}\:\mathrm{that}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{log}\left(\mathrm{sinx}\right)\mathrm{dx}=−\frac{\pi}{\mathrm{2}}\mathrm{log2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{or} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi\:\:}{\mathrm{2}}} \mathrm{log}\left(\mathrm{cosx}\right)\mathrm{dx}=−\frac{\pi}{\mathrm{2}}\mathrm{log2} \\ $$
Commented by Tinku Tara last updated on 28/Nov/17
$$\mathrm{sin}\left({x}\right)=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right) \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{x}\right) \\ $$$$\mathrm{substitute} \\ $$$${u}=\frac{\pi}{\mathrm{2}}−{x}\Rightarrow{x}=\mathrm{0},{u}=\frac{\pi}{\mathrm{2}},{x}=\frac{\pi}{\mathrm{2}},{u}=\mathrm{0} \\ $$$${du}=−{dx} \\ $$$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{x}\right)=−\int_{\pi/\mathrm{2}} ^{\mathrm{0}} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du} \\ $$$$=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du} \\ $$$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}{x}\right)\:{dx}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{x}\right)\:{dx}={I} \\ $$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \left[\mathrm{log}\:\left(\mathrm{sin}\:{x}\right)+\mathrm{log}\:\left(\mathrm{cos}\:{x}\right)\right]{dx} \\ $$$$=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{2}}{dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:\mathrm{2}{x}\right){dx}−\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{ln}\:\mathrm{2}{dx} \\ $$$$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:\mathrm{2}{x}\right){dx} \\ $$$$\mathrm{2}{x}={u}\Rightarrow{dx}={du}/\mathrm{2} \\ $$$${limits}\:\mathrm{change}\:\mathrm{to}\:\mathrm{0}\:\mathrm{to}\:\pi \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}+\int_{\pi/\mathrm{2}} ^{\pi} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[{I}+\int_{\pi/\mathrm{2}} ^{\pi} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$${t}={u}−\frac{\pi}{\mathrm{2}}\Rightarrow{u}=\frac{\pi}{\mathrm{2}}+{t}\Rightarrow\mathrm{sin}\:{u}=\mathrm{cos}\:{t} \\ $$$${dt}={du},\:\mathrm{limits}\:\mathrm{change}\:\mathrm{to}\:\mathrm{0}\:\mathrm{to}\:\frac{\pi}{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[{I}+\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{t}\right){dt}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$\mathrm{2}{I}=\frac{\mathrm{1}}{\mathrm{2}}\left[{I}+{I}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$${I}=−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$