In the rest of this unit's material we begin the study of methods of integration. There is a lot of material in this part of the course and it is important to be able to figure out when to use different methods, so make sure that you are careful in your studies of Chapter 7.
- Section 6.10: This is the official introduction of a couple of functions (ignore the inverse functions in this section). We are not going to pay sinh and cosh a lot of attention, but there are certain applications of engineering here and there that do use them. They will come back in the chapter on Taylor Series as a part of a fascinating result.
- Section 7.1: This section does no more than point out that we have substitution as our one tool for solving integrals. It points out that the substitutions can be a little tricky, which sets up sections 7.3 and 7.4, where we look carefully at two classes of interesting substitutions.
- Section 7.2: The first new method of integration, it is the only one in the chapter that is not based on clever substitutions. This one can get rather tricky, so you need to familiarize yourself with its workings. Integration by parts is, in my opinion, a highly rewarding subject - there is something fascinating in the way it solves integrals and the bootstrapping method is one of my favourite parts of calculus.
- Section 7.3: The title we are using for this section foreshadows what happens next week. There is plenty of reason to try to avoid trigonometry in the function you are integrating (the integrand) - they are often difficult to manage. In this section, we sill come up with a set of rules that will not only point out where such substitutions can work, but also which ones to choose when we have an option.