Integrating $z^n$ and $(\overline{z})^n$ along a line segment in the complex plane

Question

Let $z_1$ and $z_2$ be distinct points of $\mathbb{C}$. Let $[z_1,z_2]$ denote the oriented line segment starting at $z_1$ and ending at $z_2$. Evaluate the integral of $z^n$ and $(\overline{z})^n$ along $[z_1,z_2]$ for $n=0,1,2,\ldots$

I'm not sure where to start...

Answer 1

$f(z)=z^n$ is analytic , so, $$\int_{z_1}^{z_2}z^ndz=\frac{z^{n+1}}{n+1}|_{z_1}^{z_2}$$

But $g(z)=(\bar z)^n$ is non-analytic so, you'll need to take line integral along curve $z=tz_1+(1-t)z_2, 0\leq t\leq 1$

Thus, $$\int_{z_1}^{z_2}\bar z^ndz=(z_1-z_2)\int_{0}^{1}(t\bar z_1+(1-t)\bar z_2)^ndz$$