Question

Given the inverse Fourier transform of \[ f(s) = \begin{cases} a - |s|, & |s| \leq a \\ 0, & |s| > a \end{cases} \] The value of \[ \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx \] is:

• A. \( \pi \)
• B. \( \frac{2\pi}{3} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{4} \)

Hint:

The integral: \[ \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx \] is a well-known Fourier integral result with value \( \frac{\pi}{2} \).

Answer 1

The Correct Option is C

Step 1: Recognizing the integral. The given integral: \[ I = \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx. \] This is a standard result in Fourier analysis. 
Step 2: Evaluating the integral. Using the known result, \[ \int_0^\pi \left( \frac{\sin x}{x} \right)^2 dx = \frac{\pi}{2}. \] 
Step 3: Selecting the correct option. Since \( I = \frac{\pi}{2} \), the correct answer is (C).