The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is
The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is
$\frac{1}{2} \log \left(x^2+1+c\right)$
$\frac{1}{2} \log \left|x^2-1\right|+c$
$\log \left(x^2+1+c\right)$
$\log \left|x^2-1\right|+c$
Solution
Let $I=\int \frac{x^2-x}{x^3-x^2+x-1} d x$
$
=\int \frac{x(x-1)}{x^2(x-1)+(x-1)} d x=\int \frac{x d x}{x^2+1}
$
$
\left.=\frac{1}{2} \int \frac{2 x d x}{\left(x^2+1\right.}\right)
$
Let $x^2+1=t \Rightarrow 2 x d x=d t$
$
\begin{aligned}
& \therefore I=\frac{1}{2} \int \frac{d t}{t}=\frac{1}{2} \log t+c \\
& =\frac{1}{2} \log \left(x^2+1\right)+c
\end{aligned}
$
where ' $c$ ' is the constant of integration