How does one compute this double integral? $$\int_0^2\int_0^{y^3} x^5(2-x^{1/3})^{-1} \, dx \, dy$$
I've tried change of variables, which didn't seem to help much. I've also tried u-substitution and that didn't help. What I noticed is that there is $x=y^3$ so $y=x^{1/3}$, but I wasn't sure how to use that fact besides change of variables.

Have you tried the easiest thing? Change the order of integration.
So, let's see:
\begin{align*}
\int_0^8\int_{x^{1/3}}^2 x^5(2-x^{1/3})^{-1}\,dy\,dx &= \int_0^8 (2-x^{1/3}) x^5(2-x^{1/3})^{-1}\,dx \\ &= \int_0^2 x^5\,dx.\end{align*}
Can you finish now? 
