Fourier Sine Integral

Question

Find the Fourier sine integral of the function defined by $$ \,\mathrm{f}\left(x\right) = \begin{cases} \sin\left(x\right) & \text{;}\quad 0 \leq x \leq \pi \\[1mm] 0 & \text{;}\quad \ x\ > \pi \end{cases} $$ and hence evaluate the integral $$ \int_{0}^{\infty}{\sin\left(\pi\omega\right)\sin\left(x\omega\right) \over 1 - \omega^{2}}\,\mathrm{d}\omega\,;\quad x \geq 0 $$