Question

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

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

Answer 1

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

$\begin{array}{c}\int \frac{x^2-x}{x^3-x^2+x-1}dx\\ \Rightarrow \int \frac{x\left(x-1\right)}{x^2\left(x-1\right)+1\left(x-1\right)}dx\\ \Rightarrow \int \frac{x}{\left(x^2+1\right)}dx\\ Let{x}^2+1=t\\ \Rightarrow 2xdx=dt\\ \Rightarrow \frac{1}{2}\int \frac{dt}{t}=\frac{1}{2}\log t+c=\frac{1}{2}\log \left(x^2+1\right)+cAns.\end{array}$