I need to evaluate the following complex integral:
$ \int_{\phi}\frac{z^3}{z^2+i} dz$
where $\phi$ is the circle centered at $0$ with radius $2$
I know that there is a singularity at $z = \frac{1-i}{\sqrt{2}}$ (though I can verify this, I couldn't get this result myself either), but this singularity is inside the region so the answer is not $0$
I've considered using Cauchy's Integral Formula.
But I'm having trouble writing $\frac{z^3}{z^2+i}$ in the form $\frac{f(z)}{z-a}$
Any help would be hugely appreciated.