Answer
$$\int\frac{z^2}{z^3+1}dz=\frac{\ln|z^3+1|}{3}+C$$
Work Step by Step
$$A=\int\frac{z^2}{z^3+1}dz$$
Let $u=z^3+1$.
Then $du=3z^2dz$, so $z^2dz=\frac{1}{3}du$
Substitute into $A$, we have $$A=\frac{1}{3}\int\frac{1}{u}du$$ $$A=\frac{1}{3}\ln|u|+C$$ $$A=\frac{\ln|z^3+1|}{3}+C$$
