Integrate the function x ^ -1 x - Vidyadip

Question

Integrate the function $x \tan^{-1}x$

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Answer

Let $I = \int {x{{\tan }^{ - 1}}x} dx$$ = \int {\left( {{{\tan }^{ - 1}}x} \right).x} dx$
$= \left( {{{\tan }^{ - 1}}x} \right).\frac{{{x^2}}}{2} - \int {\frac{1}{{1 + {x^2}}}.\frac{{{x^2}}}{2}dx} $
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\int {\frac{{{x^2}}}{{1 + {x^2}}}dx} $
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\int {\frac{{{x^2} + 1 - 1}}{{{x^2} + 1}}dx}$
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\int {\left( {1 - \frac{1}{{{x^2} + 1}}} \right)dx}$
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\left( {x - {{\tan }^{ - 1}}x} \right) + c$
$= \frac{1}{2}\left[ {{x^2}{{\tan }^{ - 1}}x - x + {{\tan }^{ - 1}}x} \right] + c$
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{x}{2} + \frac{1}{2}{\tan ^{ - 1}}x + c$

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