Question

Solve: $\int ta{n}^{-1}\left(\frac{2tanx}{1-ta{n}^{2}x}\right)dx$

Answer 1

$\int ta{n}^{-1}\left[\frac{2tanx}{1-ta{n}^{2}x}\right]dx$

By application of integration by parts,

$=-\int 2xdx+xta{n}^{-1}\left(tan2x\right)$

$=-x^2+xta{n}^{-1}\left(tan\right(2x\left)\right)$

$=-x^2+xta{n}^{-1}\left(tan\right(2x\left)\right)+c$