Let $f:X→\mathbb{C}$ be a complex function which is holomorphic, where $X$ is a domain of $\mathbb{C}$. It is given that $f(z)=z^2g(z)$ for some holomorphic function $g:X→\mathbb{C}$ and $\int_{X} f(z) \mathrm{d}x=0$.
Then a question come in my mind that is $\int_{X} g(z) \mathrm{d}x=0$ also? If this integral in not zero for some $g$, then please help me to show it by some counterexample.
Take any random example of $z^2 g(z)$ and two points at which its antiderivative has the same value. There is no reason for $\int g$ to vanish along the same path.
Also: the function $1/z$ has nonzero integral over the unit circle, but the integral of $z^2(1/z)= z$ is zero along every closed path.