1. Integration by Substitution
Section 6.2a

2. substitution method of integration.
The New Method
A change of variables can often turn an
unfamiliar integral into one that we can
evaluate…
This method is called the
substitution method of integration.

3. The New Method 1. Substitute u = g(x), du = g (x)dx
Generally, this method is used when integrating a composite
function, and the derivative of the inside function is also
present in the integrand:
1. Substitute u = g(x), du = g (x)dx
2. Evaluate by finding an antiderivative F (u) of f (u)
3. Replace u by g(x)

4. Initial Practice Problems
Evaluate
Let
 Then
Substitute:
(Substitute)

5. Initial Practice Problems
Evaluate
Let
Then
Substitute:

6. Initial Practice Problems
Evaluate
Let
Then
Substitute:

7. Initial Practice Problems
Evaluate
Let
Then
Substitute:

8. Substitution in Definite Integrals
Instead of the last step we’ve been using (re-substitution???):
Substitute , and integrate with
respect to u from to

9. Evaluating Definite Integrals
Evaluate
Let
Then
Also, notice:
Substitute:

10. Two Possible Methods? Method 1 Evaluate Let Then Also, notice:
Substitute:

11. Two Possible Methods?
Method 2
Evaluate
Let
Then
Substitute:

12. Guided Practice
Evaluate the given integral.

13. Guided Practice
Evaluate the given integral.

14. Guided Practice
Evaluate the given integral.

15. Guided Practice
Evaluate the given integral.

16. Guided Practice
Evaluate the given integral.