Question

Solve $\int x{\tan }^{-1}xdx$

Answer 1

$\int x{\tan }^{-1}xdx$

${\tan }^{-1}x.\frac{x^2}{2}-\int \frac{1}{1++x^2}\times \frac{x^2}{2}dx$

$\frac{x^2}{2}{\tan }^{-1}x-\frac{1}{2}\int \frac{x^2+1-1}{1+x^2}dx$

$\frac{x^2}{2}{\tan }^{-1}x-\frac{1}{2}\int dx+\frac{1}{2}\int \frac{dx}{1+x^2}$

$\frac{x^2}{2}{\tan }^{-1}x-\frac{x}{2}+\frac{1}{2}{\tan }^{-1}x+C$