Why computing definite integrals isn't a simple matter of applying the Fundamental Theorem of Calculus, from the Wolfram Blog. There are discontinuities and such.
And this doesn't even mention integrals like that of exp(-x2) over the whole real line, for which there's a "trick", or real integrals that are best done by integrating over a contour in the complex plane -- the focus here is solely on integrals where there is a definite integral but something weird happens, the sort of thing where you think you know what you're doing but you really don't. (This is the sort of thing that teachers who have a reputation for being a bit sadistic pepper their tests with.)
(Feynman supposedly got a reputation for being really good at integration because he knew some contour integration tricks that a lot of other people didn't. He didn't know a lot of the tricks that they knew, but they only came to him after they had already banged their head against it. The moral of this story: people think you're smart if you know things that they don't. Edited: Efrique points out in a comment that I have this backwards -- but the idea still stands.)