Question

Find the integral $\int \frac{2x}{x\left(x^2+1\right)}dx$

Answer 1

Consider the given integral.

$I=\int \frac{2x}{x(x^2+1)}dx$

$I=2\int \frac{1}{(x^2+1)}dx$

We know that

$\int \frac{dx}{a^2+x^2}=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C$

Therefore,

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

Hence, this is the answer.