If 24 ∫ (sin|4x-π/12|+[2sinx])dx x ∈ to (0, π/4) =2π+α, where[•] denotes the greatest integer function, then α is equal to _______.

Question

If \(24 \int_{0}^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) d x=2 \pi+\alpha,\) where [•] denotes the greatest integer function, then \(\alpha\) is equal to _______.

Answer 1

Answer is: 12 

\(=24 \int_{0}^{\frac{\pi}{48}}-\sin \left(4 \mathrm{x}-\frac{\pi}{12}\right)+\int_{\pi / 48}^{\pi / 4} \sin \left(4 \mathrm{x}-\frac{\pi}{12}\right)\)

\(+\int_{0}^{\frac{\pi}{6}}[0] \mathrm{dx}+\int_{\pi / 6}^{\pi / 4}\left[2 \sin _{\downarrow} \mathrm{x}\right] \mathrm{dx} \)

\(= 24\left[\frac{\left(1-\cos \frac{\pi}{12}\right)}{4}-\frac{\left(-\cos \frac{\pi}{12}-1\right)}{4}\right]+\frac{\pi}{4}-\frac{\pi}{6}\)

\(=24\left(\frac{1}{2}+\frac{\pi}{12}\right)=2 \pi+12\)   

\(\alpha=12\)