If$24{\int }_0^{\frac{\mathrm{\pi }}{4}} \left(\sin \left|4\mathrm{x}-\frac{\mathrm{\pi }}{12}\right|+[2\sin \mathrm{x}]\right)\mathrm{dx}=2\mathrm{\pi }+\mathrm{\alpha }$, where [ . ] denotes the greatest integer function, then $\mathrm{\alpha }$ is equal to _________.

$\begin{array}{l}=24{\int }_0^{\frac{\mathrm{\pi }}{48}} -\sin \left(4\mathrm{x}-\frac{\mathrm{\pi }}{12}\right)+{\int }_{\mathrm{\pi }/48}^{\mathrm{\pi }/4} \sin \left(4\mathrm{x}-\frac{\mathrm{\pi }}{12}\right)\\ +{\int }_0^{\frac{\mathrm{\pi }}{6}} \left[0\right]\mathrm{dx}+{\int }_{\mathrm{\pi }/6}^{\mathrm{\pi }/4} [2\underset{1}{\downarrow }\sin \mathrm{x}]\mathrm{dx}\\ =24\left[\frac{\left(1-\cos \frac{\mathrm{\pi }}{12}\right)}{4}-\frac{\left(-\cos \frac{\mathrm{\pi }}{12}-1\right)}{4}\right]+\frac{\mathrm{\pi }}{4}-\frac{\mathrm{\pi }}{6}\\ =24\left(\frac{1}{2}+\frac{\mathrm{\pi }}{12}\right)=2\mathrm{\pi }+12\\ \mathrm{\alpha }=12\end{array}$