7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

Put $x = \tan \theta $

==> $dx = {\sec ^2}\theta \,d\theta $

$\therefore I$ $ = \int_0^{\pi /2} {\frac{{\tan \theta \,\log \,(\tan \theta )}}{{{{\sec }^4}\theta }}} {\sec ^2}\theta \,d\theta $

$ = \int_0^{\pi /2} {\sin \theta \,\cos \theta \,\log \,(\tan \theta )\,d\theta } $

$ = \frac{1}{2}\int_0^{\pi /2} {\sin 2\theta \log \,(\,\tan \theta \,)\,d\theta } $$ = 0$,

$\left\{ \because \int_{0}^{\pi /2}{\sin 2\theta \,\,\log \,\,\tan \theta \,\,d\theta =0} \right\}$.