1. Stating the problem: Given the Fourier transform problem for the function $$f(x) = \begin{cases} 1, & |x| < 1 \\ 0, & |x| > 1 \end{cases}$$ We want to find the Fourier transform $$F(\lambda) = \int_{-\infty}^{\infty} f(x) e^{-i \lambda x} dx$$. Also, we want to find $$\int_0^\infty \frac{\sin \lambda}{\lambda} d\lambda.$$ 2. Finding the Fourier transform of $$f(x)$$: Since $$f(x) = 1$$ for $$|x|<1$$ and 0 elsewhere, the integral reduces to $$F(\lambda) = \int_{-1}^1 e^{-i \lambda x} dx.$$ 3. Evaluating the integral: Using the integral of an exponential function, $$F(\lambda) = \left[ \frac{e^{-i \lambda x}}{-i \lambda} \right]_{-1}^1 = \frac{e^{-i \lambda} - e^{i \lambda}}{-i \lambda}.$$ 4. Simplify numerator using Euler's formula: $$e^{-i \lambda} - e^{i \lambda} = -2i \sin \lambda.$$ Substituting back, $$F(\lambda) = \frac{-2i \sin \lambda}{-i \lambda} = \frac{2 \sin \lambda}{\lambda}.$$ 5. Thus, the Fourier transform is $$\boxed{F(\lambda) = \frac{2 \sin \lambda}{\lambda}}.$$ 6. Now, using the Fourier transform, we find $$\int_0^\infty \frac{\sin \lambda}{\lambda} d\lambda.$$ This integral is a known integral related to the Fourier transform of the rectangle function and equals $$\frac{\pi}{2}.$$ 7. Explanation: The integral $$\int_{-\infty}^\infty \frac{\sin \lambda}{\lambda} d\lambda = \pi$$ by the Dirichlet integral. Since the integrand is even, $$\int_0^\infty \frac{\sin \lambda}{\lambda} d\lambda = \frac{\pi}{2}.$$ Final answers: $$F(\lambda) = \frac{2 \sin \lambda}{\lambda}, \quad \int_0^\infty \frac{\sin \lambda}{\lambda} d\lambda = \frac{\pi}{2}.$$