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Integration by Parts

Integration by Parts. Lesson 9.7. Review Product Rule. Recall definition of derivative of the product of two functions Now we will manipulate this to get. Manipulating the Product Rule. Now take the integral of both sides Which term above can be simplified? This gives us.

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Integration by Parts

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

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

  9. 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

  10. 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?

  11. Assignment • Lesson 9.7 • Page 396 • Exercises 1 – 25 odd, 29, 31

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