\[\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\,\mathrm{d}t=\frac{\pi}{2^{a-1}}\frac{e^{i%
\pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1)\right)},\]