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Integration by Parts

Learn to manipulate and apply the Product Rule for integration by parts efficiently through step-by-step guidelines and strategies. Practice splitting the integrand, choosing u and dv, and applying the LIATE rule for successful integration. Explore double integrals and applications for a deeper understanding.

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

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