Integration by Parts general question?

Question

In integration by parts, how do you know how many times to use the integrate by parts equation? Is it until there is no product in the integrand?

Answer 1

You stop using integration by parts when you can actually evaluate the integral (using some other method).

There are circumstances when this is not the case: if the integral you are looking for pops up when you're working with integration by parts, you may be able to solve that equation for the desired integral (a common example of this is the integral of e^x * cos(x) dx).

But in general, work toward getting an elementary integral, or at least one that can be evaluated by some technique you've already learned. http://en.wikipedia.org/wiki/Integration_by_parts has some good examples

Answer 2

Hopefully yes. Having to integrate a second time may also be indicative of not making the best choice for your U and V in UdV=UV-VdU routine.