= Solution
The <Cauchy integral formula> states that if $f$ is <holomorphic> on a neighborhood of a positively oriented simple closed contour $C$ and its interior, then for $w$ inside,
$$
f(w)=\frac1{2\pi i}\int_C\frac{f(z)}{z-w}\,dz.
$$
For the given counterclockwise circle, divide the rational function:
$$
\frac{z^3}{z^2+1}=z-\frac12\left(\frac1{z-i}+\frac1{z+i}\right).
$$
The <contour integral> of $z$ is zero because it has a primitive, while both points $\pm i$ lie inside the circle. Applying the <Cauchy integral formula> to the constant function $1$ at each point gives
$$
\boxed{\int_{|z|=2}\frac{z^3}{z^2+1}\,dz=-2\pi i}.
$$
The sign reverses if the contour orientation is reversed.
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