An integral - Riemann's derivation of number of primes

OK so I am not sure if this problem is analysis, number theory or calculus.

I am reading an English translation of Riemann's 1859 paper on the number of primes and cannot follow this bit.

He says consider this integral:

$\displaystyle \int^{+ \inf}_{+ \inf}\frac{(-x)^{s-1}dx}{e^x-1} $

From infinity in a positive sense around a domain that includes 0 but does not include any other point of discontinuity. This is then equal to:

$\displaystyle (e^{-\pi s \i}-e^{\pi s \i)\int^{+ \inf }_0 \frac{x^{s-1}dx}{e^x-1} $

I would like to fill in the details of this step but have been unsuccessful. I have tried:

Finding the residue at z=0 by solving:

$\displaystyle Res=\lim_{x \rightarrow 0} \frac{x(-x)^{s-1}}{e^x-1}=(-1)^{s-1} lim_{x \rightarrow 0}\frac{s\Pi(s-1)}{e^x}=(-1)^{s-1}s\int^{\inf}_0 \frac{x^{(s-1)}}{e^x}$

I also tried generalising x to z and integrating around the path from inf,$\displaystyle \epsilon$ to $\displaystyle -\epsilon,\epsilon$ to $\displaystyle -\epsilon,-\epsilon$ to $\displaystyle \inf,-\epsilon$ to $\displaystyle \inf,\epsilon$

where

$\displaystyle 0 < \epsilon < 2\pi$

This gave interesting results when $\displaystyle \epsilon$ either; tended to 0 equals pi or equals pi/2.

But non-the less I have not been able to get the required result. Is one of my approaches right? Is there a better approach?