# At a Glance - Lagrange (Prime) Notation

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 indefinite integrals.

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.*u*' should be similar to something else already in the 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

Evaluate the indefinite integral:

.

#### Exercise 5

What is the following indefinite integral?

#### Exercise 6

Evaluate the indefinite integral.

#### Exercise 7

Evaluate the indefinite integral.

#### Exercise 8

Find the following antiderivative.

#### Exercise 9

What is the following indefinite integral?

#### 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

Evaluate the integral.

#### Exercise 16

Integrate.

#### Exercise 17

Integrate.

#### Exercise 18

Integrate.

#### Exercise 19

Evaluate the following integral.

#### Exercise 20

Integrate.

#### Exercise 21

What's the following integral?