Integration by Parts

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. x dx ex dx ex 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

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

8. Try This • Given • Choose a uand dv • Determinethe v and the du • Substitute the values, finish integration

9. Double Trouble • Sometimes the second integral must also be done by parts

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

12. Assignment • Lesson 8.2A • Page 531 • Exercises 1 – 35 odd • Lesson 8.2B • Page 532 • Exercises 47 – 57, 99 – 105 odd