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Integration by Substitution

Integration by Substitution. Section 6.2a. The New Method. A change of variables can often turn an unfamiliar integral into one that we can evaluate…. This method is called the substitution method of integration. The New Method. Generally, this method is used when integrating a composite

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Integration by Substitution

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Presentation Transcript

  1. Integration by Substitution Section 6.2a

  2. The New Method A change of variables can often turn an unfamiliar integral into one that we can evaluate… This method is called the substitution method of integration.

  3. The New Method Generally, this method is used when integrating a composite function, and the derivative of the inside function is also present in the integrand: 1. Substitute u = g(x), du = g (x)dx 2. Evaluate by finding an antiderivativeF (u) of f (u) 3. Replace u by g(x)

  4. Initial Practice Problems Evaluate Let  Then Substitute: (Substitute)

  5. Initial Practice Problems Evaluate Then Let Substitute:

  6. Initial Practice Problems Evaluate Let Then Substitute:

  7. Initial Practice Problems Evaluate Let Then Substitute:

  8. Substitution in Definite Integrals Instead of the last step we’ve been using (re-substitution???): Substitute , and integrate with respect to u from to .

  9. Evaluating Definite Integrals Evaluate Let Then Also, notice: Substitute:

  10. Two Possible Methods? Method 1 Evaluate Then Let Also, notice: Substitute:

  11. Two Possible Methods? Method 2 Evaluate Then Let Substitute:

  12. Guided Practice Evaluate the given integral.

  13. Guided Practice Evaluate the given integral.

  14. Guided Practice Evaluate the given integral.

  15. Guided Practice Evaluate the given integral.

  16. Guided Practice Evaluate the given integral.

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