Question

$\int \sqrt{\frac{x}{{\left(x^3-1\right)}^3}}dx$

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

Answer 1

Answer: (B)

$I=\int \sqrt{\frac{x}{{\left(x^3-1\right)}^3}}dx=\int \sqrt{\frac{x}{x^9{\left(1-x^{-3}\right)}^3}}dx=\int \frac{x^{-4}}{{\left(1-x^{-3}\right)}^{3/2}}dx$
put $1-x^{-3}=t^2;3x^{-4}dx=2tdt$
$\Rightarrow x^{-4}dx=\frac{2tdt}{3}$