Параметризуем нашу окружность:

$$\{\begin{array}{l}x=a\cos t\\ y=a\sin t\end{array},0\le t\le 2\pi$$

Продифференцируем:

$$\{\begin{array}{l}dx=-a\sin tdt\\ dy=a\cos tdt\end{array}$$

$$\begin{gathered} \underset{C}{\oint }\frac{a(\cos t+\sin t)(-a\sin tdt+)-a(\cos t-\sin t)(a\cos tdt)}{a^2({\cos }^{2}t+{\sin }^{2}t)}= \\ =\underset{C}{\oint }\frac{-a^2(\cos t\sin t+{\sin }^{2}t)dt-a^2({\cos }^{2}t-\sin t\cos t)dt}{a^2({\cos }^{2}t+{\sin }^{2}t)}= \\ =-\underset{C}{\oint }1dt=-\underset{0}{\overset{2\pi }{\int }}1dt={-t|}_0^{2\pi }=-2\pi \end{gathered}$$