Thank you for reporting, we will resolve it shortly
Q.
The value of $\int \frac{d x}{x \sqrt{1-x^3}}$ is equal to
Integrals
Solution:
Put $1-x^3=t^2 \Rightarrow -3 x^2 d x=2 t d t$
$I =\int \frac{ dx }{ x \sqrt{1- x ^3}}=\int \frac{ x ^2 dx }{ x ^3 \sqrt{1- x ^3}} $
$=-\frac{1}{3} \int \frac{2 tdt }{\left(1- t ^2\right) \cdot t }=-\frac{2}{3} \int \frac{ dt }{1- t ^2}=\frac{2}{3} \int \frac{ dt }{ t ^2-1}$
$=\frac{2.1}{3.2} \ell n\left|\frac{1-t}{1+t}\right|=\frac{1}{3} \ell n\left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3+1}}\right|+c$