I'm trying to solve the following exercise of a vector field over line integral:
$$\int\limits_C\frac{-y}{4x^2+9y^2}dx+\frac{x}{4x^2+9y^2}dy,$$
where $C$ is the closed curve formed by the equations $y=x^2-1$ and $y=3$, and oriented clock-wise.
I already checked for the field to be conservative, and it isn't. I also thought in using the parametrizations $x=3\cos\theta$ and $y=2\sin\theta$, since the denominator has an elliptic shape, but it doesn't seem right... Also, I don't see that the Green theorem will make it easier than computing the integral over the curve directly.
The result is supposed to be something simple, but I don't have a clue of how to proceed, any help will be very well received :)
PS: as user10354138 pointed out, at least inside the curve I can't use the Green's theorem because of the (0,0) point.