- Topics At a Glance
- Indefinite Integrals Introduction
**Integration by Substitution: Indefinite Integrals****Legrange (Prime) Notation**- Leibniz (Fraction) Notation
- Integration by Substitution: Definite Integrals
- Integration by Parts: Indefinite Integrals
- Some Tricks
- Integration by Parts: Definite Integrals
- Integration by Partial Fractions
- Integrating Definite Integrals
- Choosing an Integration Method
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Thinking Backwards
- Improper Integrals
- Badly Behaved Limits
- Badly Behaved Functions
- Badly Behaved Everything
- Comparing Improper Integrals
- The
*p*-Test - Finite and Infinite Areas
- Comparison with Formulas
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

When we use the chain rule to take derivatives, there are some patterns that show up a lot. Some examples are

We can use these patterns to find derivatives.

The general strategy for integration by substitution has three steps:

- Change variables (substitute in
*u*for some function of*x*). - Apply an appropriate pattern to find the indefinite integral.
- Put the original variable back (substitute the function of
*x*back in for*u*).

The trickiest part is usually figuring out which function we want to replace with *u*. Here are two guidelines that might help:

*u*should be as complicated as possible, but still an "inside" function.

Example 1

Find the derivative of |

Example 2

Find . |

Example 3

Find . |

Example 4

For the integral, (a) identify |

Example 5

For the integral, (a) identify |

Example 6

For the integral, (a) identify |

Example 7

Find . |

Example 8

Find . |

Example 9

Find . |

Example 10

Find . |

Exercise 1

For the integral, (a) identify *u* and *u'* and (b) integrate by substitution.

Exercise 2

For the integral, (a) identify *u* and *u'* and (b) integrate by substitution.

Exercise 3

For the integral, (a) identify *u* and *u'* and (b) integrate by substitution.

Exercise 4

Integrate. The problem may or may not require substitution.

Exercise 5

Integrate. The problem may or may not require substitution.

Exercise 6

Integrate. The problem may or may not require substitution.

Exercise 7

Integrate. The problem may or may not require substitution.

Exercise 8

Integrate. The problem may or may not require substitution.

Exercise 9

Integrate. The problem may or may not require substitution.

Exercise 10

Integrate. The problem may or may not require substitution.

Exercise 11

Integrate. The problem may or may not require substitution.

Exercise 12

Integrate. The problem may or may not require substitution.

Exercise 13

Integrate. The problem may or may not require substitution.

Exercise 14

Integrate.

Exercise 15

Integrate.

Exercise 16

Integrate.

Exercise 17

Integrate.

Exercise 18

Integrate.

Exercise 19

Integrate.

Exercise 20

Integrate.

Exercise 21

Integrate.

Exercise 22

Integrate.