Given $F(x,y) = \langle 2xy^2+5, 2x^2y-3y^2 \rangle$, compute $\int_{C} F\cdot dr$ where $C$ is the ellipse given by $\frac{(x-4)^2}{5} + \frac{y^2}{4} = 1$.
I know that $F$ is conservative, and that $\int_{C} F\cdot dr$ is independent of path. It is just the remainder of the integral that I am confused in attempting. I converted $r(t)$ from $C$ into polar, and tried a u-sub integral with what I got from $r'(t)$, but I didn't get anywhere. I feel like I've missed something trivial. Could someone point me in the direction to start with the integration?
If $F$ is conservative, then the integral is zero. This is because $C$ is a closed curve (it begins and ends at the same point).
As you said, the integral is independent of path. If you go all the way around the ellipse, then you end at the same point you started. You could instead take a different path, which just does nothing (I mean the path just sits at that point for all values of $t$, without moving), and you would begin and end at the same point. The integral over the path that just does nothing is clearly zero.