Polynomial and closed path

Question

I need help with this task. I have to prove that $$\int_{\gamma} P(z)dz =0$$ for every polynomial $P$ and every closed path $\gamma$ in $\mathbb C$.

Answer 1

IF $\gamma$ IS PIECEWISE SMOOTH:

$\left(\dfrac{1}{n+1}a_{n}z^{n+1}+\cdots+\dfrac{1}{2}a_{1}z^{2}+a_{0}z\right)'=a_{n}z^{n}+\cdots+a_{1}z+a_{0}:=P(z)$, so the integral is just the difference of the endpoints of $y$ taken by the primitive.

Answer 2

IF $\gamma$ IS PIECEWISE SMOOTH:

Then since $\gamma$ is closed, and polynomials are entire and have primitives (which are also polynomials), we can conclude from either

1. Cauchy's Integral Theorem

2. The complex analogue of the fundamental theorem of calculus.

OTHERWISE:

Then I can understand why you would need help.