Unit 3 - Sections 7.4 to 7.6

Module Overview

This unit, we look at two major methods of integration. Like all of the methods other than IBP, these are actually just subclasses of substitutions, but they are far from obvious and deserve separate treatment. There are a lot of things happening here and plenty of example videos in which to see these details in action. Go through the material carefully and remember that you want to be able to understand when to use each method.

Learning Outcomes

• Section 7.4: In this section we plunge into a frightening realm - rather than escape trigonometry (as we did last week), we will run straight toward it. What we find is that it gives us a set of powerful tools for dealing with integrands that have quadratic terms in them and that cannot be handled using the previous methods. We have plenty of example videos to show what happens within each of these, including a very important set of examples on tangent substitutions.
• Section 7.5: The final method of integration that we learn is a way to manage any rational function we might see (or any function that, after a substitution, turns into a rational function). There are a lot of minor details here and you need to master most of them, so take your time with the example videos. At the end of this section, you will find that you can now integrate functions that seemed impossibly difficult less than a month ago. We still will not be able to integrate everything, but the class of functions that we can manage now includes a large number of the most important (and most common) functions that you will see in differential equations over the coming years.
• Section 7.6: This section mostly points out that you can have a computer do all of this work for you. Of course, we already knew that, so there is nothing new to be gained. Hopefully the video included will help you gain a little perspective on the matter, but there really is nothing else to this section.