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 e4x.

 Find .

 Find .

Example 4

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

Example 5

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

Example 6

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

 Find .

 Find .

 Find .

 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.

Integrate.

Exercise 15

Evaluate the integral.

Integrate.

Integrate.

Integrate.

Exercise 19

Evaluate the following integral.

Integrate.

Exercise 21

What's the following integral?