MCQ
$\int_{\,0}^{\,\infty } {\frac{{x\ln x\,dx}}{{{{(1 + {x^2})}^2}}}} $ is equal to
- ✓$0$
- B$1$
- C$\infty $
- DNone of these
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$,
$\left\{ \because \int_{0}^{\pi /2}{\sin 2\theta \,\,\log \,\,\tan \theta \,\,d\theta =0} \right\}$.
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