Question

Evaluate the following double integral.​

Answer 1

While the integral is doable in the given order of variables (factorize the denominator and expand into partial fractions), swapping the order make things easier.

`\displaystyle \int_0^4 \int_(\sqrt x)^2 (dy\,dx)/(4+y^3) = \int_0^2 \int_0^(y^2) (dx\,dy)/(4+y^3) \\\\ ~~~~~~~~ = \int_0^2 (y^2)/(4+y^3) \, dy \\\\ ~~~~~~~~ = \frac13 \int_4^(12) \frac{du}u \\\\ ~~~~~~~~ = \frac13 (\ln(12) - \ln(4)) = \boxed{\frac{\ln(3)}3}`