cool
\begin{align}
\oint_{|z|=2}f(z)\,dz&=\oint_{|z|=R}f(z)\,dz\\\\
&=\oint_{|z|=R} z^2\sqrt{1-\frac{1}{z^3}}\,dz\\\\
&=\oint_{|z|=R} z^2\left(1-\frac{1}{2z^3}+O\left(\frac{1}{z^6}\right)\right)\,dz\\\\
&=\int_0^{2\pi}R^2e^{i2\phi}\left(1-\frac{1}{2R^3e^{i3\phi}}+O\left(\frac{1}{R^6}\right)\right)\,iRe^{i\phi}\,d\phi\\\\
&=-i\pi+O\left(\frac1{R^3}\right)\\\\
&\to -i\pi \,\,\text{as}\,\,R\to \infty \;\;\;\;\;\; _\text{I really like this David}
\end{align}