Let $x$, $\alpha$, and $T$ be positive parameters. I'm trying to prove that the function $I(x, \alpha, T)$ defined as: $$ I(x, \alpha, T) = \frac{1}{2\pi i} \int_{\alpha-iT}^{\alpha+iT} \frac{x^s}{s} ds $$ satisfies the identity: $$ I(x, \alpha, T) = \int_{-\infty}^{\log x}e^{\alpha y}\frac{\sin Ty}{\pi y} dy. $$ I tried using the residue theorem, looking at the square (punched at the origin) of vertices $iT$, $-iT$, $\alpha-iT$ and $\alpha+iT$, but I didn't get any progress.
Thank you very much in advance.