Mathematics - Indefinite Integration Question with Solution | TestHub

Let $I=\int \frac{x^2-x}{x^3-x^2+x-1}dx$ $$=\int \frac{x(x-1)}{x^2(x-1)+(x-1)}dx=\int \frac{xdx}{x^2+1}$$ $$=\frac{1}{2}\int \frac{2xdx}{(x^2+1})$$ Let $x^2+1=t\Rightarrow 2xdx=dt$ $$\begin{array}{rl}& \therefore I=\frac{1}{2}\int \frac{dt}{t}=\frac{1}{2}\log t+c\\ & =\frac{1}{2}\log \left(x^2+1\right)+c\end{array}$$ where ' $c$ ' is the constant of integration.