Question

Evaluate: $\int \frac{x}{(x+1)(x^2+1)}dx$

Answer 1

$\begin{array}{c}{\int }^{}\frac{x}{\left(x+1\right)\left(x^2+1\right)}dx={\int }^{}\left(\frac{x+1}{2\left(x^2+1\right)}-\frac{1}{2\left(x+1\right)}\right)dx\\ =\frac{1}{2}{\int }^{}\frac{x+1}{x^2+1}dx-\frac{1}{2}{\int }^{}\frac{1}{x+1}dx\\ =\frac{1}{2}{\int }^{}\left(\frac{x}{x^2+1}+\frac{1}{x^2+1}\right)dx\\ =-\frac{\ln \left(x+1\right)}{2}+\frac{\ln \left(x^2+1\right)}{4}+\frac{{\tan }^{-1}x}{2}+C\end{array}$