Integration of Rational Functions by partial fractions

I understand how to do it but I get stuck really easily. Like this for example

$\displaystyle \int_{0}^{2\sqrt{3}} \frac{x^3}{\sqrt{16-x^2}}$

Yeah it's probably an easy one but I'm not that great with limit of integration. so I get

$\displaystyle x = 4sin(o)$

$\displaystyle dx = 4cos(o)$

So here's where I am stuck. I don't know what to do now. I plug those in but I don't know what my limit of integration is supposed to be. This stuff is killing me. I'm trying really hard to understand. Please someone help me.