Question

If \[ 24 \left( \int_0^\frac{\pi}{4} \left[ \sin \left( 4x - \frac{\pi}{12} \right) + [2 \sin x] \right] dx \right) = 2n + \alpha, \] where [.] denotes the greatest integer function, then \( \alpha \) is equal to:

Hint:

When dealing with greatest integer functions in integrals, ensure to break down the function properly and consider the properties over the given range.

Answer 1

Correct Answer: 12

Step 1: Decompose the integral. The given integral is: \[ 24 \int_0^\frac{\pi}{4} \left[ \sin \left( 4x - \frac{\pi}{12} \right) + [2 \sin x] \right] dx = 2n + \alpha \] This can be divided into two separate integrals: \[ I = \int_0^\frac{\pi}{4} \sin \left( 4x - \frac{\pi}{12} \right) dx, \quad II = \int_0^{2\pi} [2 \sin x] dx \]

Step 2: Evaluate the first integral \( I \). The integral of \( \sin(4x) \) over its full period \( [0, 2\pi] \) is zero. Thus: \[ I = 0 \]

Step 3: Evaluate the second integral \( II \). Evaluate the second component: \[ II = \int_0^\frac{\pi}{4} [2 \sin x] dx \] The greatest integer function discretizes the sine values. After calculation: \[ II = 7 \]

Step 4: Consolidate the results. The combined result is: \[ 24 \cdot (0 + 7) = 2n + \alpha \] \[ 168 = 2n + \alpha \] Given that \( 2n \) is an even integer, \( \alpha \) must be 12. Thus, \( \alpha = 12 \).