1. Integration by Parts
Lesson 9.7

2. Review Product Rule
Recall definition of derivative of the product of two functions
Now we will manipulate this to get

3. Manipulating the Product Rule
Now take the integral of both sides
Which term above can be simplified?
This gives us

4. Integration by Parts It is customary to write this using substitution
u = f(x) du = f '(x) dx
v = g(x) dv = g'(x) dx

5. Strategy Given an integral we split the integrand into two parts
First part labeled u
The other labeled dv
Guidelines for making the split
The dv always includes the dx
The dv must be integratable
v du is easier to integrate than u dv
Note: a certain amount of trial and error will happen in making this split

6. Making the Split A table to keep things organized is helpful
Decide what will be the u and the dv
This determines the du and the v
Now rewrite u v du dv x
ex dx dx ex

7. The LIATE Rule The choice for u is the first available from …
L ogarithmic
I nverse trigonometric
A lgebraic
T rigonometric
E xponential

8. Try This Given Choose a u and dv Determine the v and the du
Substitute the values, finish integration u v du dv

9. Double Trouble
Sometimes the second integral must also be done by parts u x2 v
-cos x du
2x dx dv
sin x u v du dv

10. Going in Circles When we end up with the the same as we started with
Try
Should end up with
Add the integral to both sides, divide by 2

11. What is the disk thickness?
Application
Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π
What is the volume generated by rotating the region around the y-axis?
What is the radius?
What is the disk thickness?
What are the limits?

12. Assignment Lesson 9.7A Page 396 Exercises 1, 5, 9, 13, 17, 21, 25
Lesson 9.7B
Exercises 3, 7, 11, 15, 19, 23