This is bound to come up for me in the next few days because it's one of the favorite tricks of physicsists. Legend has it that when Landau, (a legendary Russian physicist and one of my favorite authors), interviewed a new student he gave them seven problem that involved integration by parts. If they couldn't do the problems, they were out. There's a problem though. As you might have guessed from the xkcd funny strip, (picture 2), the elegance and power of the technique can serve to obfuscate a student's understanding of it, (as we often say at black-tie physics education soirees). To it more simply, I was never able to memorize or use integration by parts until I realized it can be built from the product rule of integration. Watch:
There are more pointers on how to use the formula once you have it. Basically, you've got a hammer where you're looking for an integral that's the product of a easily recognizable derivative and something else. More on how to make better use of the technique later.
1. Wikipedia on Landau
2. xkcd on integration by parts