Discontinuous complex integral I would like to compute the integral $$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+\infty}\frac{x^{s}}{s(s-1)}ds $$  for $ x \geq 1$ and $ c > 0$. I know that it should be $ \sim x $ and that if $ s-1 $ wouldn't appear at the denominator, it would be $ 1 $. Any help would be highly appreciated! 
 A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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$\ds{\Theta}$ is the Heaviside Step function. With
$\ds{x \geq 1}$ and $\ds{c > 0}$:
\begin{align}
&\color{#f00}{{1 \over 2\pi\ic}
\int_{c -\infty\ic}^{c + \infty\ic}{x^{s} \over s\pars{s - 1}}\,\dd s}
\\[3mm] = &\
\Theta\pars{c - 1}{1 \over 2\pi\ic}\,\braces{2\pi\ic\lim_{s \to 1}
\bracks{\pars{s - 1}\,{x^{s} \over s\pars{s - 1}}}} + {1 \over 2\pi\ic}\,\braces{2\pi\ic\lim_{s \to 0}
\bracks{s\,{x^{s} \over s\pars{s - 1}}}}
\\[3mm] = &\
\color{#f00}{\Theta\pars{c - 1}x - 1} =
\left\lbrace\begin{array}{lcl}
\ds{-1} & \mbox{if} & \ds{c < 1}
\\[2mm]
\ds{x - 1} & \mbox{if} & \ds{c > 1}
\end{array}\right.\,,\qquad\qquad
x \geq 1\,,\quad c > 0
\end{align}
