  Borwein integrals are one of the most popularly known phenomena in
contemporary mathematics. They were found in 2001 by David Borwein and Jonathan
Borwein and consist of a simple family of integrals involving the cardinal sine
function ``sinc'', so that the first integrals are equal to $\pi$ until,
suddenly, that pattern breaks. The classical explanation for this fact involves
Fourier Analysis techniques. In this paper, we show that it is possible to
derive an explanation for this result by means of undergraduate Complex
Analysis tools; namely, residue theory. Besides, we show that this Complex
Analysis scope allows to go a beyond the classical result when studying these
kind of integrals. Concretely, we show a new generalization for the classical
Borwein result.
