1 / 15

Integration by Substitution

Integration by Substitution. Lesson 5.5. Substitution with Indefinite Integration. This is the “backwards” version of the chain rule Recall … Then …. Substitution with Indefinite Integration. In general we look at the f (x) and “split” it into a g (u) and a du/dx So that ….

fremont
Download Presentation

Integration by Substitution

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. Integration by Substitution Lesson 5.5

  2. Substitution with Indefinite Integration • This is the “backwards” version of the chain rule • Recall … • Then …

  3. Substitution with Indefinite Integration • In general we look at the f(x) and “split” it • into a g(u) and a du/dx • So that …

  4. Substitution with Indefinite Integration • Note the parts of the integral from our example

  5. Example • Try this … • what is the g(u)? • what is the du/dx? • We have a problem … Where is the 4 which we need?

  6. Why is this now a 3? Where did the 1/3 come from? Example • We can use one of the properties of integrals • We will insert a factor of 4 inside and a factor of ¼ outside to balance the result

  7. Can You Tell? • Which one needs substitution for integration? • Go ahead and do the integration.

  8. Try Another …

  9. Assignment A • Lesson 5.5 • Page 340 • Problems:1 – 33 EOO49 – 77 EOO

  10. Change of Variables • We completely rewrite the integral in terms of u and du • Example: • So u = 2x + 3 and du = 2x dx • But we have an x in the integrand • So we solve for x in terms of u

  11. Change of Variables • We end up with • It remains to distribute the and proceed with the integration • Do not forget to "un-substitute"

  12. What About Definite Integrals • Consider a variationof integral from previous slide • One option is to change the limits • u = 3t - 1 Then when t = 1, u = 2 when t = 2, u = 5 • Resulting integral

  13. What About Definite Integrals • Also possible to "un-substitute" and use the original limits

  14. Integration of Even & Odd Functions • Recall that for an even function • The function is symmetric about the y-axis • Thus • An odd function has • The function is symmetric about the orgin • Thus

  15. Assignment B • Lesson 5.5 • Page 341 • Problems:87 - 109 EOO117 – 132 EOO

More Related