4.6 Copyright © 2014 Pearson Education, Inc. Integration Techniques: Integration by Parts OBJECTIVE Evaluate integrals using the formula for integration by parts. Solve applied problems involving integration by parts.

Slide 4- 2 Copyright © 2014 Pearson Education, Inc. THEOREM 7 The Integration-by-Parts Formula 4.6 Integration Techniques: Integration by Parts

Slide 4- 3 Copyright © 2014 Pearson Education, Inc. 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. 4.6 Integration Techniques: Integration by Parts

Slide 4- 4 Copyright © 2014 Pearson Education, Inc. 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. 4.6 Integration Techniques: Integration by Parts

Slide 4- 5 Copyright © 2014 Pearson Education, Inc. Example 1: Evaluate: 4.6 Integration Techniques: Integration by Parts

Slide 4- 6 Copyright © 2014 Pearson Education, Inc. 4.6 Integration Techniques: Integration by Parts Example 1 (concluded): Then, the Integration-by-Parts Formula gives

Slide 4- 7 Copyright © 2014 Pearson Education, Inc. 4.6 Integration Techniques: Integration by Parts Quick Check 1 Evaluate: Then, Then, the integration by parts formula gives

Slide 4- 8 Copyright © 2014 Pearson Education, Inc. 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 Attempt 2: This will not work because we do not know how to integrate 4.6 Integration Techniques: Integration by Parts

Slide 4- 9 Copyright © 2014 Pearson Education, Inc. Example 2 (continued): Attempt 3: Using the Integration-by-Parts Formula, we have This integral seems more complicated than the original. 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. Example 2 (continued): Attempt 4: Using the Integration-by-Parts Formula, we have 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. Example 3: Evaluate: Using the Integration by Parts Formula gives us 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. 4.6 Integration Techniques: Integration by Parts Quick Check 2 Evaluate: Then, Using the integration by parts formula, we get:

Slide Copyright © 2014 Pearson Education, Inc. Example 4: Evaluate: Note that we already found the indefinite integral in Example 1. Now we evaluate it from 1 to Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. Example 5: Evaluate to find the area of the shaded region shown below. 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. Example 5 (continued): Using the Integration-by-Parts Formula gives us To evaluate the integral on the right, we can apply integration by parts again, as follows. 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. Example 5 (continued): Using the Integration-by-Parts Formula again gives us Then we can substitute this solution into the formula on the last slide. 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. Example 5 (concluded): Then, we can evaluate the definite integral. 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. 4.6 Integration Techniques: Integration by Parts Quick Check 3 Evaluate: Then, Using the integration by parts formula, we get:

Slide Copyright © 2014 Pearson Education, Inc. 4.6 Integration Techniques: Integration by Parts Quick Check 3 Concluded Then we can evaluate the definite integral.

Slide Copyright © 2014 Pearson Education, Inc. Example 6: Evaluate When you try integration by parts on this problem, you will notice a pattern. 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc.

Slide Copyright © 2014 Pearson Education, Inc.

Slide Copyright © 2014 Pearson Education, Inc. f(x) and Repeated Derivatives Sign of Product g(x) and Repeated Integrals x3x3 (+)exex 3x23x2 (–)exex 6x6x(+)exex 6(–)exex 0exex Example 6 (continued): Using tabular integration can greatly simplify your work.

Slide Copyright © 2014 Pearson Education, Inc. Example 6 (concluded): Add the products along the arrows, making the alternating sign changes, to obtain 4.6 Integration Techniques: Integration by Parts

Slide Copyright © 2014 Pearson Education, Inc. 4.6 Integration Techniques: Integration by Parts Section Summary The Integration-by-Parts Formula is the reverse of the Product Rule for differentiation: The choices for u and dv should be such that the integral is simpler than the original integral. If this does not turn out to be the case, other choices should be made. Tabular integration is useful in cases where repeated integration by parts is necessary.