Integration Techniques: Substitution

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Presentation transcript:

Integration Techniques: Substitution OBJECTIVE Evaluate integrals using substitution. Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution The following formulas provide a basis for an integration technique called substitution. Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 1: For y = f (x) = x3, find dy. Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 2: For u = F(x) = x2/3, find du. Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 3: For find dy. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Example 4: For find dy. Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 5: Evaluate: Note that 3x2 is the derivative of x3. Thus, Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Quick Check 2 Evaluate: Note that Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 6: Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 7: Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Quick Check 3 Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 8: Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 9: Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Quick Check 4 Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 10: Evaluate: Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 11: Evaluate: We first find the indefinite integral and then evaluate the integral over [0, 1]. Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Example 11 (concluded): Then, we have Copyright © 2014 Pearson Education, Inc.

4.5 Integration Techniques: Substitution Section Summary Integration by substitution is the reverse of applying the Chain Rule of Differentiation. The substitution is reversed after the integration has been performed. Results should be checked using differentiation. Copyright © 2014 Pearson Education, Inc.