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Integration by Parts

Integration by Parts. Objective: To integrate problems without a u -substitution. Integration by Parts.

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Integration by Parts

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  1. Integration by Parts Objective: To integrate problems without a u-substitution

  2. Integration by Parts • When integrating the product of two functions, we often use a u-substitution to make the problem easier to integrate. Sometimes this is not possible. We need another way to solve such problems.

  3. Integration by Parts • As a first step, we will take the derivative of

  4. Integration by Parts • As a first step, we will take the derivative of

  5. Integration by Parts • As a first step, we will take the derivative of

  6. Integration by Parts • As a first step, we will take the derivative of

  7. Integration by Parts • As a first step, we will take the derivative of

  8. Integration by Parts • Now lets make some substitutions to make this easier to apply.

  9. Integration by Parts • This is the way we will look at these problems. • The two functions in the original problem we are integrating are u and dv. The first thing we will do is to choose one function for u and the other function will be dv.

  10. Example 1 • Use integration by parts to evaluate

  11. Example 1 • Use integration by parts to evaluate

  12. Example 1 • Use integration by parts to evaluate

  13. Example 1 • Use integration by parts to evaluate

  14. Guidelines • The first step in integration by parts is to choose u and dv to obtain a new integral that is easier to evaluate than the original. In general, there are no hard and fast rules for doing this; it is mainly a matter of experience that comes from lots of practice.

  15. Guidelines • There is a useful strategy that may help when choosing u and dv. When the integrand is a product of two functions from different categories in the following list , you should make u the function whose category occurs earlier in the list. • Logarithmic, Inverse Trig, Algebraic, Trig, Exponential • The acronym LIATE may help you remember the order.

  16. Guidelines • If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.

  17. Guidelines • If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.

  18. Guidelines • Since the new integral is harder than the original, we made the wrong choice.

  19. Example 2 • Use integration by parts to evaluate

  20. Example 2 • Use integration by parts to evaluate

  21. Example 2 • Use integration by parts to evaluate

  22. Example 2 • Use integration by parts to evaluate

  23. Example 3 • Use integration by parts to evaluate

  24. Example 3 • Use integration by parts to evaluate

  25. Example 3 • Use integration by parts to evaluate

  26. Example 3 • Use integration by parts to evaluate

  27. Example 4(repeated) • Use integration by parts to evaluate

  28. Example 4(repeated) • Use integration by parts to evaluate

  29. Example 4(repeated) • Use integration by parts to evaluate

  30. Example 4(repeated) • Use integration by parts to evaluate

  31. Example 4(repeated) • Use integration by parts to evaluate

  32. Example 4(repeated) • Use integration by parts to evaluate

  33. Example 5 • Use integration by parts to evaluate

  34. Example 5 • Use integration by parts to evaluate

  35. Example 5 • Use integration by parts to evaluate

  36. Example 5 • Use integration by parts to evaluate

  37. Example 5 • Use integration by parts to evaluate

  38. Example 5 • Use integration by parts to evaluate

  39. Example 5 • Use integration by parts to evaluate

  40. Example 5 • Use integration by parts to evaluate

  41. Tabular Integration • Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration.

  42. Tabular Integration • Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration. • Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column.

  43. Tabular Integration • Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration. • Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column. • Integrate f(x) repeatedly until you have the same number of terms as in the first column. List these in the second column.

  44. Tabular Integration • Integrals of the form where p(x) is a polynomial can sometimes be evaluated using a method called Tabular Integration. • Differentiate p(x) repeatedly until you obtain 0, and list the results in the first column. • Integrate f(x) repeatedly until you have the same number of terms as in the first column. List these in the second column. • Draw diagonal arrows from term n in column 1 to term n+1 in column two with alternating signs starting with +. This is your answer.

  45. Example 6 • Use tabular integration to find

  46. Example 6 • Use tabular integration to find Column 1

  47. Example 6 • Use tabular integration to find Column 1 Column 2

  48. Example 6 • Use tabular integration to find Column 1 Column 2

  49. Example 6 • Use tabular integration to find Column 1 Column 2

  50. Example 7 • Evaluate the following definite integral

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