Using Green's theorem on locally conservative field

I wish to calculate the following integral using Green's theorem:

$$\int_{\gamma}\frac{y}{x^2+y^2}dx+ \frac{-x}{x^2+y^2}dy$$

where $\gamma$ is the unit circle.

If I understand correctly, I can't use Green's theorem since the function is not $C^1$ at $(0,0)$. so the field is locally conservative but not around that point.

The idea is to use the following:

Since in the yellow area the field is conservative, the integral is $0$. the two paths that lead to the inner circle cancel out each other. The last thing I need to show is that for any field around the point $(0,0)$ the integral will be $2\pi$. How can I show that? and why does it work for every field?

• What do you mean "for any field"? That's false, of course. It is so for that specific field, but not for any ... – DonAntonio Jan 28 '18 at 10:26
• gamma isn't your boundary? – Guy Fsone Jan 28 '18 at 10:29

$$C:\;r(t):=(\cos t,\,\sin t),\,\,\,0\le t\le2\pi\implies r'(t)=(-\sin t,\,\cos t)\implies$$
$$\oint_C\frac y{x^2+y^2}dx-\frac x{x^2+y^2}dy=\int_0^{2\pi}\left(\sin t\cdot(-\sin t)-\cos t\cdot\cos t\right)dt=-\int_0^{2\pi}dt=-2\pi$$