Menu Close

please-prove-that-0-pi-2-log-sinx-dx-pi-2-log2-or-0-pi-2-log-cosx-dx-pi-2-log2-




Question Number 24852 by nnnavendu last updated on 27/Nov/17
please prove that∫_0 ^(π/2) log(sinx)dx=−(π/2)log2                                    or  ∫_0 ^((π  )/2) log(cosx)dx=−(π/2)log2
$$\mathrm{please}\:\mathrm{prove}\:\mathrm{that}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{log}\left(\mathrm{sinx}\right)\mathrm{dx}=−\frac{\pi}{\mathrm{2}}\mathrm{log2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{or} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi\:\:}{\mathrm{2}}} \mathrm{log}\left(\mathrm{cosx}\right)\mathrm{dx}=−\frac{\pi}{\mathrm{2}}\mathrm{log2} \\ $$
Commented by Tinku Tara last updated on 28/Nov/17
sin(x)=cos ((π/2)−x)  ⇒∫_0 ^( π/2) log (cos x)  substitute  u=(π/2)−x⇒x=0,u=(π/2),x=(π/2),u=0  du=−dx  ∫_0 ^( π/2) log (cos x)=−∫_(π/2) ^0 log (sin u)du  =∫_0 ^( π/2) log (sin u)du  ∫_0 ^( π/2) log (sinx) dx=∫_0 ^(π/2) log (cos x) dx=I  2I=∫_0 ^(π/2) [log (sin x)+log (cos x)]dx  =∫_0 ^( π/2) log ((sin 2x)/2)dx  =∫_0 ^(π/2) log (sin 2x)dx−∫_0 ^(π/2) ln 2dx  ∫_0 ^(π/2) log (sin 2x)dx  2x=u⇒dx=du/2  limits change to 0 to π  =(1/2)∫_0 ^π log (sin u)du−(π/2)ln 2  =(1/2)[∫_0 ^(π/2) log (sin u)du+∫_(π/2) ^π log (sin u)du]−(π/2)ln 2  =(1/2)[I+∫_(π/2) ^π log (sin u)du]−(π/2)ln 2  t=u−(π/2)⇒u=(π/2)+t⇒sin u=cos t  dt=du, limits change to 0 to (π/2)  =(1/2)[I+∫_0 ^(π/2) log (cos t)dt]−(π/2)ln 2  2I=(1/2)[I+I]−(π/2)ln 2  I=−(π/2)ln 2
$$\mathrm{sin}\left({x}\right)=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right) \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{x}\right) \\ $$$$\mathrm{substitute} \\ $$$${u}=\frac{\pi}{\mathrm{2}}−{x}\Rightarrow{x}=\mathrm{0},{u}=\frac{\pi}{\mathrm{2}},{x}=\frac{\pi}{\mathrm{2}},{u}=\mathrm{0} \\ $$$${du}=−{dx} \\ $$$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{x}\right)=−\int_{\pi/\mathrm{2}} ^{\mathrm{0}} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du} \\ $$$$=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du} \\ $$$$\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}{x}\right)\:{dx}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{x}\right)\:{dx}={I} \\ $$$$\mathrm{2}{I}=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \left[\mathrm{log}\:\left(\mathrm{sin}\:{x}\right)+\mathrm{log}\:\left(\mathrm{cos}\:{x}\right)\right]{dx} \\ $$$$=\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{log}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{2}}{dx} \\ $$$$=\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:\mathrm{2}{x}\right){dx}−\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{ln}\:\mathrm{2}{dx} \\ $$$$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:\mathrm{2}{x}\right){dx} \\ $$$$\mathrm{2}{x}={u}\Rightarrow{dx}={du}/\mathrm{2} \\ $$$${limits}\:\mathrm{change}\:\mathrm{to}\:\mathrm{0}\:\mathrm{to}\:\pi \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}+\int_{\pi/\mathrm{2}} ^{\pi} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[{I}+\int_{\pi/\mathrm{2}} ^{\pi} \mathrm{log}\:\left(\mathrm{sin}\:{u}\right){du}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$${t}={u}−\frac{\pi}{\mathrm{2}}\Rightarrow{u}=\frac{\pi}{\mathrm{2}}+{t}\Rightarrow\mathrm{sin}\:{u}=\mathrm{cos}\:{t} \\ $$$${dt}={du},\:\mathrm{limits}\:\mathrm{change}\:\mathrm{to}\:\mathrm{0}\:\mathrm{to}\:\frac{\pi}{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[{I}+\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{log}\:\left(\mathrm{cos}\:{t}\right){dt}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$$\mathrm{2}{I}=\frac{\mathrm{1}}{\mathrm{2}}\left[{I}+{I}\right]−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$$${I}=−\frac{\pi}{\mathrm{2}}\mathrm{ln}\:\mathrm{2} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *