Question

Evaluate \( \int_{0}^{\frac{3\pi}{2}} \sin\left( \left\lfloor \frac{2x}{\pi} \right\rfloor \right) dx \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function.

• A. \( \frac{\pi}{2}(\sin 1 + \cos 1) \)
• B. \( \frac{\pi}{2}(\sin 1 + \sin 2) \)
• C. \( \frac{\pi}{2}(\sin 1 - \cos 1) \)
• D. \( \frac{\pi}{2}(\sin \pi + \sin 2) \)

Hint:

For greatest integer problems, always break the integral into intervals where the expression inside the brackets is constant.

Answer 1

The Correct Option is B

Concept: The greatest integer function \( [x] \) splits the interval into subintervals where the function remains constant. We evaluate the integral piecewise.

Step 1: Find intervals where \( \left[ \frac{2x}{\pi} \right] \) is constant. \[ \frac{2x}{\pi} = \begin{cases} 0 & 0 \le x 1 & \frac{\pi}{2} \le x 2 & \pi \le x \le \frac{3\pi}{2} \end{cases} \]

Step 2: Split the integral: \[ \int_{0}^{\frac{\pi}{2}} \sin(0)\,dx + \int_{\frac{\pi}{2}}^{\pi} \sin(1)\,dx + \int_{\pi}^{\frac{3\pi}{2}} \sin(2)\,dx \]

Step 3: Evaluate each part: \[ = 0 + \sin(1)\left(\frac{\pi}{2}\right) + \sin(2)\left(\frac{\pi}{2}\right) \] \[ = \frac{\pi}{2}(\sin 1 + \sin 2) \]