Question

Integrate with respect to ${}^{\prime }{x}^{\prime }$:
$x{\tan }^{-1}x$

Answer 1

$\int x{\text{tan}}^{-1}xdx={\text{tan}}^{-1}x\int xdx-\int \frac{d\left({\text{tan}}^{-1}x\right)}{dx}(\int xdx)dx$

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

$=\frac{(x^2+1){\text{tan}}^{-1}x-x}{2}$