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Integration by Substitution

Integration by Substitution. Section 4.5. Go back and take the derivative of the “inner function”. Use power rule on u. The Mighty Chain Rule (revisited). Review: Chain rule for differentiation. If u is an expression in terms of x, then we have. For example, the general power rule is.

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Integration by Substitution

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  1. Integration by Substitution Section 4.5

  2. Go back and take the derivative of the “inner function”. Use power rule on u The Mighty Chain Rule (revisited) Review: Chain rule for differentiation If u is an expression in terms of x, then we have For example, the general power rule is

  3. Chain Rule Examples Find the derivative.

  4. We have a function (in terms of x) …and its derivative (w/respect to x) Integration So for integration we have, for F an antiderivative of f : The key to integration is recognizing the pattern.

  5. General Power Rule: Integration Rules Here are the patterns we would like to recognize: Trig Integration Rules:

  6. Examples

  7. Examples

  8. Examples

  9. Examples

  10. Examples If the integrand almost fits the pattern but is missing a constant multiple, we will employ our mathemagical manipulation powers to get theintegral in the form we need. …ok, really it’s just u-substitution and the fact that

  11. Examples

  12. Examples

  13. Examples

  14. Examples

  15. Examples

  16. Examples

  17. Examples

  18. u- substitution with a twist Example

  19. Homework Sect 4.5 page 304 # 1 – 33 odd, 43, 49, 71, 73, 87

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