Question

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

• 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)

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