$$ \int_{\gamma}\frac{dz}{z^2+1} = \frac{1}{2i}\int_{\gamma}\frac{dz}{z-i}-\frac{1}{2i}\int_{\gamma}\frac{dz}{z+i} = 2 \pi i \times0$$
By cauchy's integral formula right?
According to my lecture notes this should be $ \pi$
Where $\gamma$ is a closed contour, semicircle of radius $R$. in the upper half plane