This semester I'm teaching Calc II from Stewart's text, Calculus. We start in chapter 7, which is about techniques of integration. One of my students made a very good observation early on in the process. We had already covered u-substitution and integration by parts, and were beginning trig integrals (integrating functions that are mostly build up from trig functions, products of sines and cosines and such). He realized that the 'technique' of the section was not, exactly, a new way to integrate. It was really just a way to re-write the integrand so that you could either apply u-substitution or parts. After pointing this out in class, I realized that he was correct, and that for our purposes, there really are only two techniques of integration - u-substitution and integration by parts. Everything past that is algebraic manipulations to change how the integrand 'looks'. That isn't to say the rest of it is easy. There are plenty of strange little tricks that pop up. Odd substitutions, or strange ways to split up the function.
I've made a point of trying to convince my class that they have to go home and do lots of integrals on their own. It won't matter how many examples I do in class (where the time is limited anyway, so we can't do too many). It is always easy to watch somebody doing math and think you understand, but then get completely stuck when you sit down to do similar problems yourself. Certainly there is value in doing examples in class. For example, trig substitutions are not something most people would be likely to come up with on their own. But after an example or two, the idea is something that can be readily used, and should be struggled with individually.
What I've been doing to present examples in class is to have the class guide me, as much as possible, through a problem. I put an integral up on the board and ask for ideas. We might go through a couple of failed u-substitutions or parts attempts before striking on the right way to proceed. But that's all part of the process, and it's good to see what happens when things don't work. After we get going, I try to get the students to tell me about each step. I hope that this keeps the class involved and focused on the problem. This means the examples take more time than if I was just presenting a quick path to the answer. So I'm, perhaps, not getting to as many examples as I could be, but I think I'm happy with how it's going so far.
We've still got partial fractions to go in the chapter. I think I'm happy with how the class is going so far, and hope that feeling continues.