We'll do this example twice, once with each sort of notation. Using prime notation, take u = x v' = e^{x} Then u' = 1 and v = e^{x}. We plug all this stuff into the formula: Since the integral of e^{x} is e^{x} + C, we have We write + C instead of – C since either way we're describing the same family of functions. Using fraction notation, take u = x dv = e^{x}dx Then du = dx and v = e^{x}. We plug all this stuff into the formula: Since the integral of e^{x} is e^{x} + C, we have Much to everyone's dismay, there's no set of absolute rules for determining which function should be u. There are some guidelines, though. The whole point of integration by parts is that if you don't know how to integrate , you can apply the integrationbyparts formula to get the expression and hopefully this second integral will be easier to integrate than the original integral. If you pick u and v' incorrectly the first time, you'll probably realize it soon. Sample ProblemIf we try to integrate by parts, and we choose u = e^{x} v' = x then Putting everything into the formula, we get The new integral is which is worse than the original. This means we didn't pick u correctly! Thankfully, if we chose poorly the first time, all it means is we have to start over. When picking u and v', keep these guidelines in mind:  u ' should be simpler than u (or at least not worse!)
 v should be simpler than v' (or at least not worse!)
 You need to be able to find v from v'
 The new integral should be easier to integrate than the original integral.
