tailieunhanh - Calculus and its applications: 4.6

"Calculus and its applications: " - Integration Techniques-Integration by parts have objective: evaluate integrals using the formula for integration by parts, solve applied problems involving integration by parts. | 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Integration by Parts OBJECTIVE Evaluate integrals using the formula for integration by parts. Solve applied problems involving integration by parts. 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 7 The Integration-by-Parts Formula Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide Tips on Using Integration by Parts: 1. If you have had no success using substitution, try integration by parts. 2. Use integration by parts when an integral is of the form Match it with an integral of the form by choosing a function to be u = f (x), where f (x) can be differentiated, and the remaining factor to be dv = g (x) dx, where g (x) can be integrated. Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide 3. Find du by differentiating and v by integrating. 4. If the resulting integral is more complicated than the original, make some other choice for u = f (x) and dv = g (x) dx. 5. To check your solution, differentiate. Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Evaluate: Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Integration by Parts Example 1 (concluded): Then, the Integration-by-Parts Formula gives 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Integration by Parts Quick Check 1 Evaluate: 2012 Pearson Education, Inc. All rights reserved Slide Example 2: Evaluate: Let’s examine several choices for u and dv. Attempt 1: This will not work because we do not know how to integrate Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide Example 2 (continued): Attempt 2: . | 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Integration by Parts OBJECTIVE Evaluate integrals using the formula for integration by parts. Solve applied problems involving integration by parts. 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 7 The Integration-by-Parts Formula Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide Tips on Using Integration by Parts: 1. If you have had no success using substitution, try integration by parts. 2. Use integration by parts when an integral is of the form Match it with an integral of the form by choosing a function to be u = f (x), where f (x) can be differentiated, and the remaining factor to be dv = g (x) dx, where g (x) can be integrated. Integration Techniques: Integration by Parts 2012 Pearson Education, Inc. All rights reserved Slide 3. Find du by differentiating and v by integrating. 4. If the .