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 your 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

After class


Charles Fortin Avatar Charles Fortin
Gabriel Indurskis Avatar Gabriel Indurskis


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