Evaluate $\int_C F.ds$, with $F=(2xy\sin(x^2y)-y,\sin(x^2y))$ and $C=\{4x^2+y^2=16\}$ traversed from $(0,4)$ to $(2,0)$

Question

I'm having trouble with this one.

$F$ is not conservative and its curl is not much simpler to integrate. On the other hand, if we parametrize $C$ as $c(\theta)=(2\cos \theta, 4\sin \theta)$, we end up with embedded sines.

Any hints that would get me out of this rut?