I was given as homework to calculate the complex integral limit
$$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds $$
where $c>0$ and $k\geq1$ is an integer.
I am not so familiar with complex integration and relevant thorems (bad course choice), so Im looking for guidance on how to approach that kind of integral.
Let $\gamma$ be the counter-clockwise semi-circular path going from $c+iT$ to $c-iT$. Along $\gamma$, $|x^s| = |e^{s\log x}| = \exp(\Re(s \log x))$ is bounded by $|x^c|$, while $|s|\geq (T-c)$, hence: $$\lim_{T\to + \infty}\int_{\gamma}\frac{x^s}{s^{k+1}}ds = 0,$$ and the value of the starting integral is just the residue in $s=0$ of $\frac{x^s}{s^{k+1}}$, i.e.: $$\lim_{T\to +\infty}\frac{1}{2\pi i}\int_{c-iT}^{c+iT}\frac{x^s}{s^{k+1}}ds = \frac{\log(x)^k}{k!}.$$