Calculate $\int_\gamma \frac{1+\sin(z)}{z} dz$
where $\gamma$ is the circle of centre 0 and radius $\log(\sqrt2)$ oriented counter clockwise.
Well there is a singularity at 0. The fact that the radius is $\log(\sqrt2)$ has no real relevance as this function fails to be analytic at any circle centered at 0.
I used the Cauchy integral formula to calculate the integral and got $2\pi i$. Does that look correct?