$\int_{0}^{\frac{\pi}{2}} \sin{\left ( 2 x \right )} . \ln{\left ( \tan{x} \right )} dx$  

এর মান কত?

ধরি, $I=\int_{0}^{\frac{\pi}{2}} \sin 2 x \ln (\tan x) d x \ldots \ldots \ldots$ (i)

$\begin{array}{l} \therefore \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \sin 2\left(\frac{\pi}{2}+0-\mathrm{x}\right) \ln \left(\tan \left(\frac{\pi}{2}+0-\mathrm{x}\right)\right) \mathrm{dx}\left[\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{a}+\mathrm{b}-\mathrm{x}) \mathrm{dx}\right] \\ \therefore \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\cot \mathrm{x}) \mathrm{dx} \ldots \ldots \ldots \text { (ii) } \end{array}$

$\therefore(\mathrm{i})+(\mathrm{ii}) \Rightarrow 2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \ln (\tan \mathrm{x} \cot \mathrm{x}) \mathrm{dx} \Rightarrow 2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \sin 2 \mathrm{x} \cdot \ln 1 \mathrm{dx} \Rightarrow 2 \mathrm{I}=0 \therefore \mathrm{I}=0$