1. © 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

2. 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.

3. 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

4. 1.
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx

5. 1.
Method:
We must substitute for x and dx.
Define u as the inner function
Let
Differentiate:
Substitute for the inner function . . .
and dx

6. 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:

7. 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:

8. 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:

9. 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.

10. 3.
Do we have the derivative of the inside “lurking around”?

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

12. So,
where
Integrate:
Substitute back:

13. 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.

14. Exercises 5. 4.
Give exact answers.

15. 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

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

17. 5.
Let
Limits:
So,
We can use the log laws to simplify this.

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