Question

If \( [x] \) is the greatest integer \( \le x \), then \( \pi^2 \int_{0}^{2} \left( \sin \frac{\pi x}{2} \right) (x - [x])^{[x]} dx \) is equal to :

• A. \( 2(\pi + 1) \)
• B. \( 2(\pi - 1) \)
• C. \( 4(\pi + 1) \)
• D. \( 4(\pi - 1) \)

Hint:

Always split integrals involving Greatest Integer Function \( [x] \) or Fractional Part Function \( \{x\} \) at integer boundary points.
Remember \( (x-[x])^0 = 1 \) for all \( x \) except integers where it might be undefined, but for integration, we care about the interval interior.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to split the integral into two parts where the behavior of \( [x] \) is constant: \( [0, 1) \) and \( [1, 2) \).
Step 2: Key Formula or Approach:
For \( x \in [0, 1) \), \( [x] = 0 \).
For \( x \in [1, 2) \), \( [x] = 1 \).
Step 3: Detailed Explanation:
Let \( I = \int_{0}^{2} \left( \sin \frac{\pi x}{2} \right) (x - [x])^{[x]} dx \).
Split the integral:
\[ I = \int_{0}^{1} \sin\left(\frac{\pi x}{2}\right) (x - 0)^0 dx + \int_{1}^{2} \sin\left(\frac{\pi x}{2}\right) (x - 1)^1 dx \]
\[ I_1 = \int_{0}^{1} \sin\left(\frac{\pi x}{2}\right) dx = \left[ -\frac{2}{\pi} \cos\left(\frac{\pi x}{2}\right) \right]_0^1 = -\frac{2}{\pi}(0 - 1) = \frac{2}{\pi} \]
\[ I_2 = \int_{1}^{2} (x-1) \sin\left(\frac{\pi x}{2}\right) dx \]
Use Integration by Parts (ILATE) where \( u = x-1, v = \sin(\pi x/2) \):
\[ I_2 = \left[ (x-1) \left(-\frac{2}{\pi} \cos\frac{\pi x}{2}\right) \right]_1^2 - \int_{1}^{2} 1 \cdot \left(-\frac{2}{\pi} \cos\frac{\pi x}{2}\right) dx \]
\[ I_2 = \left[ 1 \cdot (-\frac{2}{\pi} (-1)) - 0 \right] + \frac{2}{\pi} \left[ \frac{2}{\pi} \sin\frac{\pi x}{2} \right]_1^2 \]
\[ I_2 = \frac{2}{\pi} + \frac{4}{\pi^2} (\sin \pi - \sin \frac{\pi}{2}) = \frac{2}{\pi} + \frac{4}{\pi^2} (0 - 1) = \frac{2}{\pi} - \frac{4}{\pi^2} \]
Total integral \( I = I_1 + I_2 = \frac{2}{\pi} + \frac{2}{\pi} - \frac{4}{\pi^2} = \frac{4}{\pi} - \frac{4}{\pi^2} \).
We need \( \pi^2 I \):
\[ \pi^2 \left( \frac{4\pi - 4}{\pi^2} \right) = 4(\pi - 1) \]
Step 4: Final Answer:
The result is \( 4(\pi - 1) \).