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