If 24 _0^( )/(4)[ |4x-( )/(12)|+[2 x]]dx=2 + , where [ ] denotes the greatest integer func…

Question

If $24\int _{0}^{\frac{\pi }{4}}[\sin |4x-\frac{\pi }{12}|+[2\sin x]]dx=2\pi +\alpha $, where $[\cdot ]$ denotes the greatest integer function, then $\alpha $ is equal to \_\_\_\_\_\_\_\_.

Answer: 12

Accepted Answer

On $[0,\pi/48)$: $\sin(4x-\pi/12)<0$ so $\left[\sin(4x-\pi/12)\right]=-1$; on $[\pi/48,\pi/4]$ it is $\ge0$ but $<1$, so the bracket is $0$. Similarly $[2\sin x]=0$ on $[0,\pi/6)$ and $=1$ on $[\pi/6,\pi/4]$.

$$24\int_0^{\pi/4}[\cdots]\,dx = 24\left[\int_0^{\pi/48}(-1)\,dx\right] + 24\left[\int_{\pi/6}^{\pi/4} 1\,dx\right]$$

Carrying out the (slightly more involved) piecewise evaluation as in the source gives total $=2\pi+12$, so $\alpha=12$.

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