int_{0}^{pi /2} {sin 2x log tan x , dx} is eq | vedclass

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(C) Let $I = \int_{0}^{\pi /2} {\sin 2x \log 	an x \, dx}$.Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_

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(C) Let $I = \int_{0}^{\pi /2} {\sin 2x \log 	an x \, dx}$.Using the property $\int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx$,we get:$I = \int_{0}^{\pi /2} {\sin 2(\frac{\pi}{2} - x) \log 	an(\frac{\pi}{2} - x) \, dx}$$I = \int_{0}^{\pi /2} {\sin(\pi - 2x) \log \cot x \, dx}$Since $\sin(\pi - 2x) = \sin 2x$ and $\log \cot x = \log(	an x)^{-1} = -\log 	an x$,we have:$I = \int_{0}^{\pi /2} {\sin 2x (-\log 	an x) \, dx}$$I = - \int_{0}^{\pi /2} {\sin 2x \log 	an x \, dx}$$I = -I$$2I = 0 \Rightarrow I = 0$.

$\int_{0}^{\pi /2} {\sin 2x \log \tan x \, dx}$ is equal to

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