$\displaystyle L\left\{\frac{1}{t}\sin(\alpha t)\right\} = \int_0^\infty e^{-st}\frac{1}{t}\sin(\alpha t) dt$
Do integration by parts with the two functions $\displaystyle u(t)=e^{-st}$ and $\displaystyle v'(t)=\frac{\sin(\alpha t)}{t}$
You could also try this: rewrite the above integral as $\displaystyle \int_0^\infty e^{-st}\frac{1}{t}e^{i\alpha t} dt$, solve and take the imaginary part of your answer.