Recently, I had an exam and in that I was asked to evaluate the line integral of the function $F=\frac{-y}{x^2+y^2}i+\frac{x}{x^2+y^2}j$ alongside the unit circle, $0 \le t \le 2\pi $ . Moreover, it was asked if this integral could be carried out with using Green's Theorem or not and why?
For the first, I did the following: $\oint_C F\cdot dr=\int_0^{2\pi}F\big (\cos(t),\sin(t)\big)\cdot\big(-\sin(t),\cos(t)\big)dt$ which is $2\pi$. But always I have problem with above theorem and I do know I didn't pass this part of question correctly. May I ask to help me? Thank you.