Question

$\underset{0}{\overset{\sqrt{3}}{\int }}{\tan }^{-1}\left(\frac{2x}{1+x^2}\right)dx$

Answer 1

${\int }_0^{\sqrt{3}}{\tan }^{-1}\left(\frac{2x}{1+x^2}\right)dx$ $x=\tan \theta \Rightarrow dx={\sec }^2\theta d\theta$

$={\int }_0^{\pi /3}2\theta {\sec }^2\theta d\theta$

$\Rightarrow {\int }_0^{\pi /3}x\tan x+\ln |\cos \left(x\right)|=\frac{\pi }{3}\sqrt{3}+\ln \frac{1}{2}\Rightarrow \frac{\pi }{\sqrt{3}}-\ln 2$