Let $x>0$. Calculate:
$$\lim_{R\rightarrow\infty} \int_{\frac{1}{2}-iR}^{\frac{1}{2}+iR}\frac{x^s}{s}ds$$
for $x=1$ it is quite simple. Otherwise, I used the following contours:
The left one when $x>1$ and the right one when $0<x<1$. Let us discuss the case $x>1$.
The integral on $\Gamma_1$ is what we want. The integral on $\Gamma_3$ vanishes since:
$$\bigg| \int_{\Gamma_3}f(s)\bigg|\leq \int_{\Gamma_3}|f(s)| \leq 2R \bigg|\frac{x^s}{s}\bigg| \leq 2R \frac{e^{\ln x \Re s}}{\sqrt{2}R}=\sqrt{2}e^ {-R\ln x} \rightarrow 0$$
But I have no idea what to do with $\Gamma_2$ and $\Gamma_4$.