The author tells the story of typical undergraduate instruction for the integrals of tan(x) and sec(x). I would have thought that such a setting would have included that these were improper integrals, because of the infinities in the functions. i.e. if you evaluate the definite integral for any particular interval, it will give you the right answer except if you've gone through the part where the graph goes up to plus infinity and back through minus infinity.

Can someone more mathematically literate than me shed any light on whether it matters? I guess it's still useful even if the integral is undefined at certain points off the edges of the map.