let $S= \{(x,y) | x^2 +y^2 >0 \}$ and let $$P(x,y) = \frac{y}{x^2+y^2} , Q(x,y) = \frac{-x}{x^2+y^2}$$ where $(x,y) \in S $. Let $C$ be a piecewise smooth Jordan curve lying in $S$
If $(0,0)$ is inside $C$ then find the line integral $\int_{C} Pdx + Qdy $
My attempt : By using the green theorem $$\int_{C} Pdx + Qdy =\int \int_{R}( \frac {dq}{dx} - \frac{dp}{dy}) dxdy= \int \int_{R} \frac{y^2-x^2}{(x^2 +y^2)^2 } -\frac{y^2-x^2}{(x^2 +y^2)^2 }dxdy=0$$
Is its true ?
Hint:
Draw a little circle around $0$ inside of $C$. Connect that circle to $C$ by a simple curve (a line segment is generally sufficient). The region between the circle, $C$, and the curve is simply connected.