Evaluate $$\int_{\Gamma} \frac{1}{z^2+1}$$ where $\Gamma=\{z:|z|=2\}$
One way is to use partial fractions and then cauchy integral formula.
The other way is to look at:
$$\int_{\Gamma} \frac{1}{z^2+1}=\int_{\gamma_1}\frac{\frac{1}{z-i}}{z+i}+\int_{\gamma_2}\frac{\frac{1}{z+i}}{z-i}$$
Where $\gamma_1$ is a curve around $i$ and $\gamma_2$ is a curve around $-i$ but in which direction? clockwise? as $\Gamma$ is anti clockwise?