This is a question found in Theodore Gamelin's Complex Analysis, Chapter 3, Section 2.
We are given the differential $\frac{-ydx+xdy}{x^2+y^2}\quad \text{where }\ (x,y)\neq(0,0)$ The first part says to show that the differential is closed, which I've already done.
The second part says to show that the line integral in any annulus centered at $0$ is not independent of path. I need help for this second part.
I tried evaluating the line integral over the piecewise boundaries $|z|=R$ and $|z|=r$. When calculating the latter, I put a negative sign since the inner circle should have the opposite orientation. But both integrals came out to be $2\pi$ and hence the sum equaled $0$.