Question

Solve : $\int \frac{\left(x{\tan }^{-1}x\right)}{(1+x^2{)}^{\frac{3}{2}}}dx$

Answer 1

$I=\int \frac{\left(x{\tan }^{-1}x\right)}{(1+x^2{)}^{\frac{3}{2}}}dx$

Let ${\tan }^{-1}x=\theta \Rightarrow d\theta =\frac{1}{1+x^2}dx$

$\Rightarrow x=\tan \theta$

Now,

$I=\int \frac{\tan \theta .\theta }{\sqrt{1+{\tan }^2\theta }}.d\theta$

$=\int \frac{\theta \tan \theta }{\sec \theta }d\theta$

$=\int \theta \sin \theta d\theta$

$=-\theta \cos \theta +\int \cos \theta d\theta$

$=-\theta \cos \theta +\sin \theta +c$

$=\sin \left({\tan }^{-1}x\right)-{\tan }^{-1}x\cos \left({\tan }^{-1}x\right)+c$