1 / 54

Mastering Integration by Parts: A Comprehensive Guide to Evaluating Integrals

This guide focuses on the technique of integration by parts, essential for integrating products of functions without a u-substitution. It outlines step-by-step guidelines for choosing appropriate functions, u and dv, and provides numerous examples to practice this technique. It emphasizes the importance of experience in finding the best choices for integration to simplify the process. With the helpful acronym LIATE to remember the order of function categories, this resource is designed for students looking to enhance their integral calculus skills.

brent
Télécharger la présentation

Mastering Integration by Parts: A Comprehensive Guide to Evaluating Integrals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Warm-up: Evaluate the integrals. 1) 2)

  2. Warm-up: Evaluate the integrals. 1) 2)

  3. Warm-up: Evaluate the integrals. 1) 2)

  4. Integration by Parts Section 8.2 Objective: To integrate problems without a u-substitution

  5. 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.

  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 • As a first step, we will take the derivative of

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

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

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

  12. 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.

  13. Example 1 • Use integration by parts to evaluate

  14. Example 1 • Use integration by parts to evaluate

  15. Example 1 • Use integration by parts to evaluate

  16. Example 1 • Use integration by parts to evaluate

  17. Example 1 • Use integration by parts to evaluate

  18. 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.

  19. 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.

  20. 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.

  21. 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.

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

  23. Example 2 • Use integration by parts to evaluate

  24. Example 2 • Use integration by parts to evaluate

  25. Example 2 • Use integration by parts to evaluate

  26. Example 2 • Use integration by parts to evaluate

  27. Example 2 • Use integration by parts to evaluate

  28. Example 3 (S): • Use integration by parts to evaluate

  29. Example 3 • Use integration by parts to evaluate

  30. Example 3 • Use integration by parts to evaluate

  31. Example 3 • Use integration by parts to evaluate

  32. Example 3 • Use integration by parts to evaluate

  33. Example 4 (Repeated): • Use integration by parts to evaluate

  34. Example 4 (Repeated): • Use integration by parts to evaluate

  35. Example 4 (Repeated): • Use integration by parts to evaluate

  36. Example 4 (Repeated): • Use integration by parts to evaluate

  37. Example 4 (Repeated): • Use integration by parts to evaluate

  38. Example 4 (Repeated): • Use integration by parts to evaluate

  39. Example 4 (Repeated): • Use integration by parts to evaluate

  40. Example 4 (Repeated): • Use integration by parts to evaluate

  41. Example 5: • Evaluate the following definite integral

  42. Example 5: • Evaluate the following definite integral

  43. Example 5: • Evaluate the following definite integral

  44. Example 5: • Evaluate the following definite integral

  45. Example 5: • Evaluate the following definite integral

  46. Example 5: • Evaluate the following definite integral

  47. Example 5: • Evaluate the following definite integral

  48. Example 5: • Evaluate the following definite integral

  49. Example 5: • Evaluate the following definite integral

  50. Example 5: • Evaluate the following definite integral

More Related