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