Business Calculus presentation on integration

Various types of integration useful in business calculus. Focuses primarily on a wide use of integrals and examples of the integration by parts formula

1. Calculus for Business, Economics, Life Sciences, and Social Sciences Fourteenth Edition Chapter 6 Additional Integration Topics Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

2. Section 6.3 Integration by Parts

3. Integration by Parts (1 of 2) In the indefinite integral none of the integration techniques previously considered can be used to find an antiderivative for l n x . The technique integration by parts, discussed in this section, will enable finding and many other integrals. Integration by parts is based on the product formula for derivatives which states, if f and g are differentiable functions, then

4. Integration by Parts (2 of 2) Integrating both sides of the derivative of a product formula gives The first integral to the right of the equal sign is This gives This integral equation can be transformed to a more convenient form if

5. Integration-by-Parts Formula The integral formula can be transformed to a more convenient form if Integration-by-Parts Formula

6. Example 1: Integration by Parts Find using integration by parts. Check the result. Solution Write the integration-by-parts formula: With Substitute into

7. Example 1: Integration by Parts-Check Results With

8. Example 2: Integration by Parts Find Solution Let With

9. Summary Summary Integration by Parts: Selection of For The product u d v must equal the original integrand . It must be possible to integrate d v . The new integral should not be more complicated than the original integral. For integrals involving For integrals involving

10. Example 3: Repeated Use of Integration by Parts (1 of 4) Find Solution Choose Substitute into

11. Example 3: Repeated Use of Integration by Parts (2 of 4) Apply the integration-by-parts formula to the second integral choosing Then

12. Example 3: Repeated Use of Integration by Parts (3 of 4) Substitute this result into the previous integral, we have

13. Example 3: Repeated Use of Integration by Parts (4 of 4) Check

14. Example 4: Using Integration by Parts (1 of 2) Find and interpret the result geometrically. Solution Choose

15. Example 4: Using Integration by Parts (2 of 2) With the result the graphical interpretation is that the area under the curve of This is illustrated by the shaded region shown in Figure 1 .