How does integration by parts work?

The method of integration by parts is working upon the Product Rule in differentiation. We know the Product Rule to be d/dt (uv) = uv' + vu', where u and v are separate functions and u' and v' are the corresponding differentiated functions. Using this, we can get I[d/dt (uv)dt] = I[(uv' + vu')dt], where I[] is the integral of the functions within the square brackets. This then gives us uv = I[uv' dt] + I[vu' dt]. This can then be arranged to give the formula of Integration by Parts which is I[uv' dt] = uv - I[vu' dt]. This is how you derive the formula of Integration by Parts, however you will not be expected to know this in your exams, this is simply a way of helping you to understand where the formula comes from and to put away any confusion you may have had on this topic.