MCQ
$\int_{ - 1}^1 {x{{\tan }^{ - 1}}x\,dx} $ equals
- ✓$\left( {\frac{\pi }{2} - 1} \right)$
- B$\left( {\frac{\pi }{2} + 1} \right)$
- C$(\pi - 1)$
- D$0$
$\because \,\,\,x{{\tan }^{-1}}x$ is an even function
$I = [2\frac{{{x^2}}}{2}{\tan ^{ - 1}}x]_0^1 - 2\int_0^1 {\frac{1}{2}\frac{{{x^2}}}{{1 + {x^2}}}dx} $
$I = [{x^2}{\tan ^{ - 1}}x]_0^1 - \int_0^1 {\frac{{{x^2} + 1 - 1}}{{1 + {x^2}}}dx} $
$I = [{x^2}{\tan ^{ - 1}}x]_0^1 - [x]_0^1 + [{\tan ^{ - 1}}x]_0^1$
==> $I = \frac{\pi }{4} - 1 + \frac{\pi }{4} = \frac{\pi }{2} - 1$.
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