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