hi MathsIsFun,

I've always taught this as

Of course it comes to the same thing, but it avoids having a double integral in the rule, so it looks easier to take in.

Also you can easily 'prove' it from the product rule of differentiation.

For choosing u and v I say look for a 'u' that gets easier/simpler when you differentiate it and a 'dv/dx' that doesn't get any more complicated when you integrate it.  You could mention this when you do the e^x times x example.

Good examples .... I especially liked the one that just gets more complicated ... good because it will happen to every student at some time and this shows what to do about it.

Also: would it be better to give the first example BEFORE saying "to help you remember" and the steps?

Your first example is a good starter.  If you adopt my suggestion (i) below, then you could pose the problem, introduce what u and dv/dx might be and show how the rule can provide a solution.  I would suggest that you also differentiate the answer to demonstrate that it worked.  (And include a comment that doing that is a test for any integral).  Then you are reday to state the rule formally.

So, suggestions

(i) consider the alternative formulation of the rule.

(ii) maybe include after some examples a 'why does it work' bit.

(iii) add this example

After one application of the rule the problem gets no simpler.

Apply it again and you get back minus the original integral!

But, now re-arrange to make the original once more the subject and you have the solution.  I just loved that example when I first met it.

Bob

OK, posted a new version:  Integration by Parts (use refresh!)

Slightly rearranged, with Bob's example at the end, and a footnote on its relation to Product Rule.

What do you all think now?