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: Break the integral into two parts. The given integral is: \[ 24 \int_0^\frac{\pi}{4} \left[ \sin \left( 4x - \frac{\pi}{12} \right) + [2 \sin x] \right] dx = 2n + \alpha \] We can split this into two 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: Solve for the first integral \( I \). The integral of \( \sin(4x) \) over the interval from 0 to \( 2\pi \) will cancel out (since it's a complete period of the sine function). So, we have: \[ I = 0 \]

Step 3: Solve for the second integral \( II \). Now, evaluate the second part of the integral: \[ II = \int_0^\frac{\pi}{4} [2 \sin x] dx \] The greatest integer function will split the sine values into intervals where it holds constant values. After evaluating, we find: \[ II = 7 \]

Step 4: Combine the results. Now, we have: \[ 24 \cdot (0 + 7) = 2n + \alpha \] \[ 168 = 2n + \alpha \] Given that \( 2n \) is an integer multiple of 2, we find that \( \alpha = 12 \). Thus, \( \alpha = 12 \).