Contour integration of logarithmic function I have the following problem:

Consider the function $$f(z)=\ln(z+1)-\ln(z-1)$$
  This has branch points at $z=1$ and $z=-1$, and we place the branch cut joining the two branch points along the real line. Consider the contour segments about the branch points:  A: Anticlockwise arc of infinitesimal radius, encircling $z=1$ from just below the cut to just above.
B: Anticlockwise arc of infinitesimal radius encircling $z=-1$ from just above the cut to just below.
Calculate the integral $\int f(z) dz$ over the contours A and B. 

My attempt:
I parameterize segment A as such:
$$ A \equiv \{z(\theta)=1+\epsilon e^{i\theta} \,\vert\, \theta\in(-\pi,\pi)\} $$
Now computing the integral (choosing the principal branch):
$$\begin{align} \int_Af(z)dz &= \lim_{\epsilon\to 0}\int_{-\pi}^\pi f(z(\theta))\,z'(\theta)\,d\theta \\
&= \lim_{\epsilon\to 0}\int_{-\pi}^\pi \ln \bigg\vert\frac{\epsilon e^{i\theta}+2}{\epsilon e^{i\theta}}\bigg\vert \, i\epsilon e^{i\theta}\,d\theta \\
&= \lim_{\epsilon\to 0}\int_{-\pi}^\pi \ln \bigg\vert 1+\frac{2}{\epsilon e^{i\theta}}\bigg\vert \, i\epsilon e^{i\theta}\,d\theta
\end{align}$$
At which point I'm stuck on how to evaluate the integral. I want to check if I have parameterized A correctly, and that it is just a matter of finding a way to plough through the integration; Or whether there is a smarter way of parameterizing/setting up the integral. 
I only require assistance on A, as B will be very similar once I get an idea of how to compute the integral. Thanks all!
