Answer
$$0$$
Work Step by Step
The curve can be parameterized as: $$x= \cos \theta; y= \sin \theta$$ and $$ 0 \leq \theta \lt 2 \pi $$
and $$dx= -\sin \theta d \theta; \\ dy=\cos \theta d \theta$$
We need to set up the line integral and compute the integrand of the double integral as follows:
$\int_{C} F \cdot dr= -\int_{S} \dfrac{2xy}{(x^2+y^2)^2} dx+\int_{S} \dfrac{y^2-x^2}{(x^2+y^2)^2} dy$
or, $=-\int_{S} \dfrac{2(\cos \theta)(\sin \theta)}{((\cos \theta)^2+(\sin \theta)^2)^2} (-\sin \theta d \theta )+\int_{S} \dfrac{(\sin \theta)^2-(\cos \theta)^2}{((\cos \theta)^2+(\sin \theta)^2)^2} (\cos \theta d \theta ) \\=\int_{0}^{-2 \pi} [\cos 2 \theta \cos \theta +\sin 2 \theta \sin \theta ] d \theta \\=\int_{0}^{-2 \pi} \cos \theta d\theta \\=[\sin \theta ]_0^{-2 \pi}\\ =\sin (-2 \pi)-\sin 0 \\=0$
