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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 ….

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Integration by Substitution

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  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

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