Show or prove that $\int_{-\infty}^{\infty}\frac{\sin(x)}{x} \mathrm{e}^{i \alpha x} \mathrm{d}x = \pi$

$\sin(x)/x$ is roughly the Fourier transform of a box function, see http://en.wikipedia.org/wiki/Rectangular_function#Fourier_transform_of_the_rectangular_function

Then using the convolution theorem this integral is just the convolution of a box function (of width $\pi$ and height $1$) with the constant $1$ function, which is the result.