Handling a complex line integral (Homework) 
Evaluate $$\oint_{\gamma} \frac{\bar{z}}{8+z}dz$$ where $\gamma$ is the rectangle whose endpoints are $\pm 3 \pm i$, oriented counterclockwise. 

This one has been a relatively frustrating struggle for me. My attempts have centered around breaking the rectangle into the four obvious pieces, e.g. $\gamma_1 = (3-6t)+i$, $0 \leq t \leq 1$ etc. Then, I've tried direct substitution, e.g.: $$\int_{\gamma_1} \frac{\bar{z}}{8+z}dz=\int_0^1 \frac {(3-6t)-i}{11-6t+i}(-6)\,dt$$
and I've also tried multiplying the numerator and denominator by $8+\bar{z}$ and then converting to real and imaginary components. Either way, I'm stuck on horrific computations and I keep wondering if there is not some simpler hint I am failing to see. 
 A: Well, the first thing I see is that your integral is of the form
$$\int \frac{\text{polynomial in }t}{\text{other polynomial in }t}\mathrm{d}t$$
In other words, it's the integral of a rational function. These are in some sense the "simplest" functions to integrate; not in the sense that the integral will always be easy to figure out, but there is a more or less straightforward procedure for doing them using partial fraction decomposition, so you're at least guaranteed to get an answer, which is more than you can say for an arbitrary function.
In this particular case, both the polynomials are linear, so it's a particularly easy case. You could try the partial fraction method, or you could check your textbook/notes or search online for information on how to do this type of integral specifically. Remember, it's basically this:
$$\int\frac{x - a}{x - b}\mathrm{d}x$$
That doesn't look so bad, does it?
A: Well, to make the computations easier 
I would say first make the representation a little bit easier by $$I=\oint \frac{\bar{z}+z-z}{8+z} \, dz =\oint \frac{2\text{Re}(z)}{8+z} \, dz-\oint \frac{z}{z+8} \, dz   $$
Since the branch point is away from the contour we can find an analytic branch of the logarithm where the branch cut is chosen away from the contour.
