Mellin transform of sin(x) .

Question

Proof of the identity $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$ for $\alpha\in (0,2)$.

In the above post, in the 2nd answer (given by mark viola) - in the 2nd part - "ALTERNATIVE METHODOLOGY: CONTOUR INTEGRATION" equation no 8 goes to zero as $\varepsilon\to0^+$ only if $\alpha <1$ but we need to prove the identity for $\alpha\in (0,2)$. Is the proof incomplete then? If it's incomplete then how to extend the proof upto $\alpha < 2$.