So, my wife is finishing up a graduate degree, and is thus currently in a course called "Mathematical Methods for Physicists". They're covering a heck of a lot of ground, but one of her recent homework questions caused me great offense:

∫(0,∞) sin(bx) dx

"A-ha," I thought, "the partial sums do not converge, therefore this integral is divergent." But the actual answer was different:

How the heck did that work? Why hasn't someone outlawed this? How can we find an integral for something that diverges like that? (The use of Euler I understand, it's "how can you suddenly add this new variable in there")