If $ 24 \int\limits_0^{\frac{\pi}{4}} \bigg[\sin \left| 4x - \frac{\pi}{12} \right| + [2 \sin x] \bigg] dx = 2\pi + \alpha $, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________.
Answer (integer)
12
Solution
<p>$$\begin{aligned}
= & 24 \int_0^{\frac{\pi}{48}}-\sin \left(4 \mathrm{x}-\frac{\pi}{12}\right)+\int_{\pi / 48}^{\pi / 4} \sin \left(4 \mathrm{x}-\frac{\pi}{12}\right) \\
& +\int_0^{\frac{\pi}{6}}[0] \mathrm{dx}+\int_{\pi / 6}^{\pi / 4}[2 \sin \mathrm{x}] \mathrm{dx}
\end{aligned}$$</p>
<p>$$\begin{aligned}
& =24\left[\frac{\left(1-\cos \frac{\pi}{12}\right)}{4}-\frac{\left(-\cos \frac{\pi}{12}-1\right)}{4}\right]+\frac{\pi}{4}-\frac{\pi}{6} \\
& =24\left(\frac{1}{2}+\frac{\pi}{12}\right)=2 \pi+12 \\
& \alpha=12
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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