Download presentation
Presentation is loading. Please wait.
1
Techniques of Integration
Chapter 9 Techniques of Integration Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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 Copyright © 2014, 2010, 2007 Pearson Education, Inc.
3
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 9.2 Integration by Parts Copyright © 2014, 2010, 2007 Pearson Education, Inc.
4
Integration by Parts Using Integration by Parts
Section Outline Integration by Parts Using Integration by Parts Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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). Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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 Copyright © 2014, 2010, 2007 Pearson Education, Inc.
7
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts CONTINUED Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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 Copyright © 2014, 2010, 2007 Pearson Education, Inc.
11
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Using Integration by Parts CONTINUED Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.