7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$I = \int_0^{\pi /2} {\sin 2\,\left( {\frac{\pi }{2} - x} \right)\log \tan \left( {\frac{\pi }{2} - x} \right)\,\,dx} $,

$[\because \int_{0}^{a}{f\,(x)\,dx=\int_{0}^{a}{f\,(a-x)\,dx]}}$

$ = \int_0^{\pi /2} {\,\,\,\,\,\sin 2x\log \cot x\,\,dx} $

$ = - \int_0^{\pi /2} {\,\,\,\,\,\sin 2x\log \tan x\,\,dx} $

$\therefore I = - I\,\,==> 2I = 0$ $ \Rightarrow I = 0.$