Question

Find the integral:

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

Answer 1

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

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

$=\frac{1}{2}\int \frac{dx}{x+1}-\frac{1}{2}\int \frac{xdx}{x^2+1}+\int \frac{dx}{x^2+1}$

$=\frac{1}{2}\log \left(x+1\right)-\frac{1}{2}\times \frac{1}{2}\log \left(x^2+1\right)+\frac{1}{2}{\tan }^{-1}x+c.$

$=\frac{1}{2}\log \left(x+1\right)-\frac{1}{4}\log \left(x^2+1\right)+\frac{1}{2}{\tan }^{-1}x+c$