Question

The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. x is

• A. $\frac{1}{2}log\left(x^2+1+c\right)$
• B. $\frac{1}{2}log\left|x^2-1\right|+c$
• C. $log\left(x^2+1+c\right)$
• D. $log\left|x^2-1\right|+c$

Answer 1

Answer: (A)

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