© Adapted from Christine Crisp Integration by Substitution 5 Examples After you copy the 5 examples into your notebook and watch the video clip, you may take Part 2 of the Quiz starting at 1:10pm!!! Good Luck! © Adapted from Christine Crisp

Integration by substitution can be used for a variety of integrals: some compound functions, some products and some quotients. Sometimes we have a choice of method.

1. Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . Find dx by treating like a fraction

1. Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx

1. Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx

1. Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx Integrate: Replace u:

1. Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx Integrate: Replace u:

1. Method: We must substitute for x and dx. Define u as the inner function Let Differentiate: Substitute for the inner function . . . and dx Integrate: Replace u:

The “Thinking Method” 2. We have already met this type of integral. See if you can do it. Solution: Reversing the Chain Rule gives This is an example of a problem that could be done both ways.

3. Do we have the derivative of the inside “lurking around”?

Cancel the extra x 3. Define u as the inner function: Let Differentiate: Substitute for the inner function and dx Cancel the extra x

So, where Integrate: Substitute back:

Definite integration We work in exactly the same way BUT we must also substitute for the limits, since they are values of x and we are changing the variable to u. A definite integral gives a value so we never return to x.

Exercises 5. 4. Give exact answers.

4. Solutions: Let Limits: You can change the limits and keep “u” or “u-sub” back to “x” once you integrate & keep the original limits. “x” limits “u” limits

Solutions: 4. Let Limits: So You can change the limits and keep “u” or “u-sub” back to “x” & keep the original limits. So

5. Let Limits: So, We can use the log laws to simplify this.

Monday’s Homework – Textbook 6.2 # 18, 22, 24, 32, 40, 44, 54, 56, 62, 66