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

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

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

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

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

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

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

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

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

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

4.6 Integration Techniques: Integration by Parts
Example 4: Evaluate: Note that we already found the indefinite integral in Example 1. Now we evaluate it from 1 to 2. Copyright © 2014 Pearson Education, Inc.

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

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

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

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

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

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