Question

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

Answer 1

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

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

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

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

$=\frac{1}{2}\ln (x-1)-\frac{1}{2}\left[\frac{\ln (x^2+1)}{2}+{\tan }^{-1}\left(x\right)\right]$

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