Evaluate : \[\int_{0}^{\frac{\pi}{2}}[\log (\sin \mathrm{x})-\log (2 \cos \mathrm{x})] d \mathrm{x} \]

$$\text { Let } \mathrm{I}=\int_{0}^{\pi / 2}[\log \sin \mathrm{x}-\log (2 \cos \mathrm{x})] \mathrm{dx}=\int_{0}^{\pi / 2} \log \left(\frac{\tan \mathrm{x}}{2}\right) \mathrm{dx} Using property $\displaystyle \int_{0}^{a} f(\mathrm{x}) d \mathrm{x}=\int_{0}^{a} f(a-\mathrm{x}) d \mathrm{x}$ We get, $\displaystyle \mathrm{I}=\int_{0}^{\pi / 2} \log \left(\frac{\cot \mathrm{x}}{2}\right) \mathrm{dx}$ $\displaystyle \therefore 2 \mathrm{I}=\int_{0}^{\pi / 2} \log \left(\frac{\tan \mathrm{x}}{2} \times \frac{\cot \mathrm{x}}{2}\right) \mathrm{dx}=\int_{0}^{\pi / 2} \log \left(\frac{1}{4}\right) \mathrm{dx}$ $\displaystyle 2 \mathrm{I}=\left.\log \left(\frac{1}{4}\right) \mathrm{x}\right|_{0} ^{\pi / 2}=\frac{\pi}{2} \log \frac{1}{4}$ \mathrm{I}=\frac{\pi}{4} \log \frac{1}{4} \mathrm{OR}-\frac{\pi}{2} \log 2 $$