1. Integration by Substitution
Lesson 5.5

2. Substitution with Indefinite Integration
This is the “backwards” version of the chain rule
Recall …
Then …

3. Substitution with Indefinite Integration
In general we look at the f(x) and “split” it
into a g(u) and a du/dx
So that …

4. Substitution with Indefinite Integration
Note the parts of the integral from our example

5. Where is the 4 which we need?
Example
Try this …
what is the g(u)?
what is the du/dx?
We have a problem …
Where is the 4 which we need?

6. Where did the 1/3 come from?
Example
We can use one of the properties of integrals
We will insert a factor of 4 inside and a factor of ¼ outside to balance the result
Where did the 1/3 come from?
Why is this now a 3?

7. Can You Tell? Which one needs substitution for integration?
Go ahead and do the integration.

8. Try Another …

9. Assignment A
Lesson 5.5
Page 340
Problems: 1 – 33 EOO 49 – 77 EOO

10. Change of Variables
We completely rewrite the integral in terms of u and du
Example:
So u = 2x and du = 2 dx
But we have an x in the integrand
So we solve for x in terms of u

11. Change of Variables We end up with
It remains to distribute the and proceed with the integration
Do not forget to "un-substitute"

12. What About Definite Integrals
Consider a variation of integral from previous slide
One option is to change the limits
u = 3t Then when t = 1, u = when t = 2, u = 5
Resulting integral

13. What About Definite Integrals
Also possible to "un-substitute" and use the original limits

14. Integration of Even & Odd Functions
Recall that for an even function
The function is symmetric about the y-axis
Thus
An odd function has
The function is symmetric about the orgin

15. Assignment B
Lesson 5.5
Page 341
Problems: EOO 117 – 132 EOO