Integrals — Class 12 Maths Solution

Let$I = \int\limits_0^{\pi /2} {\left( {2\log \,sin\,x - log\,sin\,2x} \right)} dx$

$= \int\limits_0^{\pi /2} {\left[ {2\log \,sin\,x - log\,\left( {2sin\,x\cos x} \right)} \right]} dx$

$= \int\limits_0^{\pi /2} {\left[ {2\log \,sin\,x - log2\, - \log \sin x - \log \cos x} \right]} dx$

$= \int\limits_0^{\pi /2} {\log \sin xdx - \left( {\log 2} \right)} \int\limits_0^{\pi /2} {\left( 1 \right)dx} - \int\limits_0^{\pi /2} {\log \cos \left( {\cfrac{\pi }{2} - x} \right)dx}$

$= \int\limits_0^{\pi /2} {\log \sin xdx - \left( {\log 2} \right)\left[ x \right]_0^{\pi /2} - \int\limits_0^{\pi /2} {\log {\mathop{\rm sinx}\nolimits} \,dx} }$

$= - \left( {\log 2} \right)\left( {\cfrac{\pi }{2} - 0} \right) = - \cfrac{\pi }{2}\log 2 = \cfrac{\pi }{2}\log {\left( 2 \right)^{ - 1}} = \cfrac{\pi }{2}\log \left( {\cfrac{1}{2}} \right)$