Evaluate:
$\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$.
$\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$
$\int_C (z-z_0)^{n-1}dz $ for any integer $n$, where C is the contour once around the circle $|z-z_0|=1$.
I know that $z=x+iy$, so $\hat{z}=x-iy$. so for the first part, $\int (x-iy)dz$ from $i$ to $2-i$, but I don't know if to integrate with respect to $dx$ or $dy$?
What you are missing is the description of the contour on which you are integrating. For example, the straight line from $z=i$ to $z=2-i$ is described by
$$z=i (1-t) + (2-i) t$$
where $t \in [0,1]$. Then equating real and imaginary parts, $x=2 t$, $y=1-2 t$. Also, $dz = 2 (1-i) dt$. The first integral is then
$$\int_0^1 (2 t - i (1-2 t)) (2 (1-i)) \, dt$$
You can evaluate this.