The given question asks to evaluate the integral:

$\int_0^\infty \left( \frac{x \cos x - \sin x}{x^3} \right) \cos \frac{x}{2} \, dx$

using the Fourier integral representation of the function ( f(x) = \begin{cases} 1 - x^2, & |x| < 1 \ 0, & |x| > 1 \end{cases} ).

Approach:

This is a standard problem that can be solved using the Fourier transform of $f(x)$. The function $f(x)$ is a simple polynomial within the interval $[-1, 1]$, and its Fourier transform will be helpful in solving this integral. Here's the general outline:

1. Fourier Transform: Use the Fourier integral representation of $f(x)$ to express the integral in a form that can be compared with standard results.
2. Break Down the Integral: Split the given expression $\frac{x \cos x - \sin x}{x^3}$ and compute the Fourier transform.

After solving the integral using standard Fourier techniques or lookup tables, the value of the integral turns out to be:

$\boxed{\frac{3\pi}{8}}$

Answer: $\frac{3\pi}{8}$

Would you like further details on the steps involved in solving the integral? Also, here are some related questions to extend this topic:

1. What is the Fourier transform of a rectangular function?
2. How does the Fourier transform of a polynomial behave?
3. Can we solve this integral using residue calculus or other methods?
4. How does the Parseval's theorem help in evaluating integrals?
5. What are the applications of Fourier transforms in solving differential equations?

Tip: Fourier transform techniques can greatly simplify the evaluation of integrals involving oscillatory functions!