Parametrize the contours of integration

Question

I am having a difficult time figuring this problem out:

Parametrize the contours of integration and write the integrals in terms of the parametrizations.

$$\int_{\Gamma} (3\bar{z}^2+2z^3)\,dz$$

where $\Gamma$ is a straight line segment that joins $1-2i$ (initial point) and $3+4i$ (terminal point).

I would appreciate any help anyone can offer.

Answer 1

The straight line from $1-2i$ to $3+4i$ is given by the parameterization $\gamma(t)=(1-2i)(1-t)+(3+4i)t = 1+2t +i(6t-2)$ where $t \in [0,1]$
We have $dz=(2+6i)dt$
What does the integral come down to? Fill in the placeholders: $$ \int_0^1(3(\cdot)^2+2(\cdot)^3)(\cdot)dt $$