1. Substitution
Lesson 7.2

2. Review Recall the chain rule for derivatives
We can use the concept in reverse
To find the antiderivatives or integrals of complicated formulas
We look for integrands that fit the right side of the chain rule

3. Strategy We look for an expression that can be the "inside" function
We substitute u = g(x)
We also determine what is du or g'(x)

4. Integration by Substitution
Now we have
Then we use the general power rule for integrals
Finally substitute u = x2 + 1 back in

5. Substitution Method
We seek the following situations where we can substitute u in as the "inner" function
Let u represent the quantity under a root or raised to a power
Let u represent the exponent on e
Let u represent the quantity in the denominator

6. Example Consider the problem of taking the integral of
Strategy … substitute u = 4x – 6
What is the derivative of u with respect to x?
Now we make the substitution
The ¼ adjusts for the 4 in the du

7. Substitution The resulting integral is much simpler
Now we reverse the substitution and simplify

8. Try Another What will we substitute … u = ? What is the du ?
Now rewrite the integral and proceed

9. How About Another? Consider u = ? du = ? u = x2 + 5 du = 2x dx
Problem …
2x is not a constant
Cannot adjust the integral with a constant coefficient
Substitution will not work for this integral

10. Indefinite Integral of u -1
If it looked like this we could do it
u = x du = 2x dx
Then use rule for integral of u -1
Final result:

11. Indefinite Integral of eu
Try this:
What is the u? the du?
u = x du = 4x3 dx
Rewrite, adjust for the factor of 4 in the du

12. Practice
Try these

13. Application
We are told that a certain bacteria population is increasing a rate of
What is the increase in the population during the first 8 hours

14. Assignment Lesson 7.2A Page 449 Exercises 3 – 41 odd Lesson 7.2B
Exercises 39 – 44 all