Overview
We now begin our study of more advanced integration techniques. Recall that the (first) Fundamental Theorem of Calculus says that we can evaluate a definite integral \(\int_{a}^{b} f(x) \; dx\) if we know an antiderivative \(F(x)\) of \(f(x)\). Our goal for the rest of Chapter 5 is to study some ways to find antiderivatives. Much like derivative “rules”, this is a list of techniques and shortcuts.
In this section, we see how it is possible to revert the chain rule. This technique is called integration by substitution. Knowing how to antidifferentiate basic functions and the substitution rule will allow us to antidifferentiate more complex functions.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.

Recognize the notation for an indefinite integral and state its meaning.

Given a composite function whose “inner” function is linear, find its general antiderivative.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Use proper notation when using Integration by Substitution. In particular, fully convert an integral from \(x\)’s to \(u\)’s and back again, without mixing the two variables.

Use Integration by Substitution in unusual cases, such as those where there is not an obvious substitution.
To prepare for class

Do the first part of the Preview Activity for this section (on WeBWorK if required by your teacher).

Read subsection 5.3.1: Reversing the Chain Rule: First Steps.

Watch the following quick recap video:

Do the remaining parts of the Preview Activity for this section (on WeBWorK if required by your teacher).

Read subsection 5.3.2: Reversing the Chain Rule: usubstitution.

Watch at least two of any of the following videos. These videos give examples on how to apply the substitution method and you can always come back to them later and watch them all if you are still having trouble applying that technique once we will have worked on it in class.
Note: Many of these examples are of the form \(u=kx\), where \(k\) is some constant, and most of their antiderivatives can in fact often be found much faster by reverse engineering! See these as good practice problems, but in the long run, you should aim for using substitution only for more complicated cases.

Watch this video about using Substitution directly for definite integrals: