1. Integration Techniques: Integration by Parts
OBJECTIVE
Evaluate integrals using the formula for integration by parts.
Solve applied problems involving integration by parts.
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2. 4.6 Integration Techniques: Integration by Parts
THEOREM 7
The Integration-by-Parts Formula
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3. 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.
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4. 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.
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5. 4.6 Integration Techniques: Integration by Parts
Example 1: Evaluate:
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6. 4.6 Integration Techniques: Integration by Parts
Example 1 (concluded):
Then, the Integration-by-Parts Formula gives
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7. 4.6 Integration Techniques: Integration by Parts
Quick Check 1
Evaluate:
Then,
Then, the integration by parts formula gives
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8. 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
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9. 4.6 Integration Techniques: Integration by Parts
Example 2 (continued):
Attempt 2:
Using the Integration-by-Parts Formula, we have
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10. 4.6 Integration Techniques: Integration by Parts
Example 3: Evaluate:
Using the Integration by Parts Formula gives us
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11. 4.6 Integration Techniques: Integration by Parts
Quick Check 2
Evaluate:
Then,
Using the integration by parts formula, we get:
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12. 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.
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13. 4.6 Integration Techniques: Integration by Parts
Example 5: Evaluate to find the area of the shaded region shown below.
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14. 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.
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15. 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.
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16. 4.6 Integration Techniques: Integration by Parts
Example 5 (concluded):
Then, we can evaluate the definite integral.
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17. 4.6 Integration Techniques: Integration by Parts
Quick Check 3
Evaluate:
Then,
Using the integration by parts formula, we get:
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18. 4.6 Integration Techniques: Integration by Parts
Quick Check 3 Concluded
Then we can evaluate the definite integral.
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19. 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.
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