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 $π$ 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.