$\int\limits_0^{\pi / 2} \sin 2 x \log \tan x d x$ is equal to

0

Let $I=\int\limits_0^{\pi / 2} \sin 2 x \log \tan x d x$           .......(i)

Then,

$I=\int\limits_0^{\pi / 2} \sin 2\left(\frac{\pi}{2}-x\right) \log \tan \left(\frac{\pi}{2}-x\right) d x$

$\Rightarrow I =\int\limits_0^{\pi / 2} \sin 2 x \log \cot x d x$             ....(ii)

Adding (i) and (ii), we get

$2 I=\int\limits_0^{\pi / 2} \sin 2 x(\log \tan x+\log \cot x) d x$

$\Rightarrow 2 I=\int\limits_0^{\pi / 2} \sin 2 x \times \log 1 d x \Rightarrow 2 I=0 \Rightarrow I=0$