Answer
$0$
Work Step by Step
The parameterize representation for the curve is:
$x= \cos \theta; y= \sin \theta$ and $ 0 \leq \theta \lt 2 \pi $ and $dx= -\sin \theta d \theta$ and $dy=\cos \theta d \theta$
We need to work out with the line integral and evaluate the integrand of the double integral as follows:
$\int_{C} F \cdot dr= -\int_{S} \dfrac{2xy}{(x^2+y^2)^2} dx+\int_{S} \dfrac{y^2-x^2}{(x^2+y^2)^2} dy$
$=-\int_{S} \dfrac{2(\cos \theta) \times (\sin \theta)}{((\cos \theta)^2+(\sin \theta)^2)^2} \times (-\sin \theta d \theta )+\int_{S} \dfrac{(\sin \theta)^2-(\cos \theta)^2}{((\cos \theta)^2+(\sin \theta)^2)^2} (\cos \theta d \theta )$
$=\int_{0}^{-2 \pi} [\cos 2 \theta \cos \theta +\sin 2 \theta \sin \theta ] d \theta $
$=\int_{0}^{-2 \pi} \cos \theta d\theta $
$=[\sin \theta ]_0^{-2 \pi}$
$=0$
