Evaluate: ∫_0^π / 2(2 log cos x- log sin 2 x) d x Case-Study Based Question

\(\displaystyle \int_{0}^{\frac{\pi}{2}}(2 \log \cos x-\log \sin 2 x) d x\) \[\begin{aligned} & =\int_{0}^{\frac{\pi}{2}} \log \left(\frac{\cos ^{2} x}{2 \sin x \cos x}\right) d x \\ & =\int_{0}^{\frac{\pi}{2}} \log \left(\frac{\cot x}{2}\right) d x=I \text { (say) } \end{aligned} \] \[I=\int_{0}^{\frac{\pi}{2}} \log \frac{\tan x}{2} d x \quad \text { (using property) } \] Adding ($\displaystyle 1$) and \[2 I=\int_{0}^{\frac{\pi}{2}} \log \left(\frac{\tan x}{2} \frac{\cot x}{2}\right) d x \] \[2 I=\log \left(\frac{1}{4}\right)[x]_{0}^{\pi / 2} \] \[I=\frac{\pi}{4} \log \frac{1}{4} \quad \text { or } \quad-\frac{\pi}{2} \log 2 \]