If 24 0 4 ( sin | 4 x – 12 | + [ 2 sin x ] ) d x = 2, where [ · ] denotes the greatest integer function, then is equal to _________. [2025]

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Solution

Q.
If $24 \int_{0}^{\frac{π}{4}} (\sin | 4 x - \frac{π}{12} | + [2 \sin x]) d x = 2 π + α$ , where $[\cdot]$ denotes the greatest integer function, then $α$ is equal to _________. [2025]

Ans.
(12)

Let $I = 24 \int_{0}^{\frac{π}{4}} (\sin | 4 x - \frac{π}{12} | + [2 \sin x]) d x$

$= 24 [\int_{0}^{π / 48} [- \sin (4 x - \frac{π}{12})] d x + \int_{π / 48}^{π / 4} \sin (4 x - \frac{π}{12}) d x + \int_{0}^{π / 6} [0] d x + \int_{π / 6}^{π / 4} [1] d x]$

$= 24 [\frac{(1 - \cos \frac{π}{12}) - (- 1 - \cos \frac{π}{12}) + \frac{π}{4} - \frac{π}{6}}{4}]$

$= 24 (\frac{1}{2} + \frac{π}{12}) = 12 + 2 π$

$\Rightarrow I = 12 + 2 π = 2 π + α$ [Given]

On comparing, we get $α = 12$ .

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