Question

Evaluate $\int \frac{x^2}{x^3-1}dx.$

• A. $\frac{\ln |x^3-1|}{3}$
• B. $\frac{\ln |x^3+1|}{3}$
• C. $\frac{\ln |x^3+1|}{3}+3x$
• D. $\frac{\ln |x^3+1|}{3}+\frac{3x}{2}$

Answer 1

The correct option is A $\frac{\ln |x^3-1|}{3}$

Let $I=\int \frac{x^2}{x^3-1}dx$

Substitute $x^3-1=t$

$\Rightarrow 3x^{2}dx=dt$

$\Rightarrow x^{2}dx=\frac{1}{3}dt$

So, $I=\int \frac{1}{t}\cdot \frac{1}{3}dt=\frac{1}{3}\int \frac{dt}{t}=\frac{1}{3}\ln |t|+C=\frac{\ln |x^3-1|}{3}+C$