Question

The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ with respect to $x$ is
(where $C$ is constant of integration)

• A. $\frac{1}{2}\ln |x^2+1|+C$
• B. $\frac{1}{2}\ln |x^2-1|+C$
• C. $\ln |x^2+1|+C$
• D. $\ln |x^2-1|+C$

Answer 1

The correct option is A $\frac{1}{2}\ln |x^2+1|+C$

Let $I=\int \frac{x^2-x}{x^3-x^2+x-1}dx$
$\Rightarrow I=\int \frac{x(x-1)}{x^2(x-1)+(x-1)}dx$
$\Rightarrow I=\int \frac{xdx}{x^2+1}$
$\Rightarrow I=\frac{1}{2}\int \frac{2xdx}{(x^2+1)}$
Let $x^2+1=t$
$\Rightarrow 2xdx=dt$
$\therefore I=\frac{1}{2}\int \frac{dt}{t}$
$\Rightarrow I=\frac{1}{2}\ln |t|+C$
$\therefore I=\frac{1}{2}\ln |x^2+1|+C$