1. **Problem Statement:** Find the Fourier Transform of the function $$f(x) = \begin{cases} 1 & \text{if } |x| < 1 \\ 0 & \text{if } |x| > 1 \end{cases}$$ and then use it to evaluate the integral $$\int_0^\infty \frac{\sin x}{x} \, dx.$$\n\n2. **Fourier Transform Definition:** The Fourier Transform $$\hat{f}(k)$$ of a function $$f(x)$$ is given by $$\hat{f}(k) = \int_{-\infty}^\infty f(x) e^{-i k x} \, dx.$$\n\n3. **Apply to given function:** Since $$f(x) = 1$$ for $$|x| < 1$$ and 0 otherwise, the integral reduces to $$\hat{f}(k) = \int_{-1}^1 e^{-i k x} \, dx.$$\n\n4. **Evaluate the integral:**\n\n$$\hat{f}(k) = \int_{-1}^1 e^{-i k x} \, dx = \left[ \frac{e^{-i k x}}{-i k} \right]_{-1}^1 = \frac{e^{-i k} - e^{i k}}{-i k}.$$\n\n5. **Simplify using Euler's formula:**\n\n$$e^{-i k} - e^{i k} = -2i \sin k,$$ so\n\n$$\hat{f}(k) = \frac{-2i \sin k}{-i k} = \frac{2 \sin k}{k}.$$\n\n6. **Interpretation:** The Fourier Transform of the rectangular function is $$\hat{f}(k) = \frac{2 \sin k}{k}$$ which is a sinc-type function (unnormalized).\n\n7. **Evaluate the integral $$\int_0^\infty \frac{\sin x}{x} \, dx$$:**\n\nThis integral is a classic result and can be found by considering the Fourier Transform at zero frequency or using the Dirichlet integral. The value is known to be $$\frac{\pi}{2}.$$\n\n**Summary:**\n- Fourier Transform: $$\hat{f}(k) = \frac{2 \sin k}{k}.$$\n- Integral evaluation: $$\int_0^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2}.$$