1. Techniques of Integration
Chapter 9
Techniques of Integration
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2. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter Outline
Integration by Substitution
Integration by Parts
Evaluation of Definite Integrals
Approximation of Definite Integrals
Some Applications of the Integral
Improper Integrals
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3. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 9.2
Integration by Parts
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4. Integration by Parts Using Integration by Parts
Section Outline
Integration by Parts
Using Integration by Parts
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5. Integration by Parts
The following equation is the principle of integration by parts and is one of the most important techniques of integration.
G(x) is an antiderivative of g(x).
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6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts
EXAMPLE
Evaluate.
SOLUTION
Our calculations can be set up as follows:
Differentiate
Integrate
Then
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7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts
CONTINUED
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8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts
EXAMPLE
Evaluate.
SOLUTION
Our calculations can be set up as follows:
Then
Notice that the resultant integral cannot yet be solved using conventional methods. Therefore, we will attempt to use integration by parts again.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts
CONTINUED
Our calculations can be set up as follows:
Then
Therefore, we have
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts
EXAMPLE
Evaluate.
SOLUTION
Our calculations can be set up as follows:
Then
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11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts
CONTINUED
Copyright © 2014, 2010, 2007 Pearson Education, Inc.