Question

Evaluate $\int {\tan }^{-1}xdx$.

Answer 1

We have to find the value of $\int {\tan }^{-1}xdx$

In order to find this, we use the formula $\int f^{-1}\left(x\right)dx=x{f}^{-1}\left(x\right)-\int f\left(y\right)dy$ , where $y=f^{-1}\left(x\right)$

By the above formula, we get

$\int {\tan }^{-1}xdx=x{\tan }^{-1}x-\int \tan ydy$ , where $y={\tan }^{-1}\left(x\right)$

$⟹\int {\tan }^{-1}xdx=x{\tan }^{-1}x-\ln |\cos y|+c$

$⟹\int {\tan }^{-1}xdx=x{\tan }^{-1}x-\ln |\cos \left({\tan }^{-1}x\right)|+c$

$⟹\int {\tan }^{-1}xdx=x{\tan }^{-1}x+\frac{1}{2}\ln (1+x^2)+c$