1. Integration by Parts
Lesson 8.2

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 du dv v x
ex dx dx ex

7. Strategy Hint Trick is to select the correct function for u
A rule of thumb is the LIATE hierarchy rule The u should be first available from
Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential

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

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

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. Assignment
Lesson 8.2A
Page 531
Exercises 1 – 35 odd

12. dv and its anti-derivatives
Tabular Method
Consider
Determine u, dv as before
Set up table
Quit when you get 0 in middle column
Solution obtained by adding signed products of diagonal entries
Alternating Signs
u and its derivatives
dv and its anti-derivatives +
sin 4x - 2x 2

13. 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?

14. Found this in your teacher's waste basket.
Where Is the Error?
Found this in your teacher's waste basket.
He's indefinitely nuts!

15. Assignment
Lesson 8.2B
Page 532
Exercises 47 – 57 odd, 99, 101, 107