Q:

Stuck on Complex Contour Integration

I don't know where to begin with this question. Can anyone help me?
Let  be a semicircular path with centre at  and with radius of 1 in the left half plane, that is, it is the straight line from 2 to 0 followed by left half a circle from 0 back to 2i. 

Evaluate $$\oint_C \bar z  dz$$

A:

hint 
Along the vertical line, we have
$$z=it$$ with
$$0\le t\le 2, \bar{z}=-it, \;\; dz=idt$$
along the half-circle
$$z-i=e^{it}$$
with
$$\frac{\pi}{2}\le t\le \frac{3\pi}{2}$$

