Here's the following integral:
$$\oint\limits_C\frac{(x+y)\;\mathrm{d}x-(x-y)\;\mathrm{d}y}{x^2+y^2}$$
$C$ is given by $x^2+y^2=a^2$
Actually I do not know how to approach it, I tried to parametrize it but it did not work properly, I have no more ideas.
If you parametrise as $x=a\cos \theta$ and $y=a\sin \theta$, then your integral reduces to $$ \frac{a^2}{a^2} \int_0^{2\pi} \left((\cos \theta +\sin \theta)(-\sin \theta) -(\cos \theta-\sin \theta)(\cos \theta)\right)d\theta$$ $$=-\int_0^{2\pi} \left(\cos^2 \theta +\sin^2 \theta\right)d\theta$$ $$=-2\pi$$
Hope this helps you.