Answer
$$0$$
Work Step by Step
Green's Theorem states that:
$\oint_C A\,dx+B \,dy=\iint_{D}(\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y}) dA $
We need to set up the line integral as follows:
$\int_{C} F \cdot dr= -\int_{C} \dfrac{y}{x^2+y^2} i+\int_{C} \dfrac{x}{x^2+y^2} j (dx i +dyj)=-\int_{C} \dfrac{y}{x^2+y^2} dx+\int_{C} \dfrac{x}{x^2+y^2} dy$
Then we have:
$$\int_{C} F \cdot dr =\oint_C-\dfrac{y}{x^2+y^2} dx+\int_{C} \dfrac{x}{x^2+y^2} dy=\iint_{D}(\dfrac{\partial (\dfrac{x}{x^2+y^2} )}{\partial x}-\dfrac{\partial (-\dfrac{y}{x^2+y^2} )}{\partial y})dA\\=\iint_{D} \dfrac{y^2-x^2}{(x^2+y^2)^2}-\dfrac{y^2-x^2}{(x^2+y^2)^2} \\=0$$
