Chapter 7 Additional Integration Topics Section 4 Integration Using Tables

2 Learning Objectives for Section 7.4 Integration Using Tables The student will be able to ■ Use a table of integrals. ■ Use substitution and reduction formulas. ■ Solve application problems.

3 Using a Table of Integrals Table II of Appendix C contains integral formulas illustrating some basic integrations. More extensive tables are available for other integrals.

4 Example Table II, formula 27 fits: with u = x, du = dx, a = 16 and b = 1.

5 Substitution and Integral Tables Sometimes the formula matches exactly, as in the preceding example. Sometimes a substitution needs to be made in order to fit one of the formulas on the table.

6 Example This almost fits formula 41: If u = 3x, u 2 = 9x 2, du = 3dx and a = 1, we could make the necessary adjustments.

7 Example (continued)

8 Reduction Formulas Sometimes using the table will not solve the integral directly, but instead replaces the given integral with one that has an exponent reduced by 1. This type of formula is called a reduction formula and means we need to apply the formula in the table repeatedly until the integral is completely evaluated.

9 Example Formula 47 fits: This last integration was done by parts! First use: Second use: Third use:

10 Application: Producers’ Surplus Find the producers’ surplus at a price level of $20 for the price- supply equation Step 1. Find, the supply when the price is $20

11 Application (continued) Step 2. Sketch a graph: Step 3. Find the producers’ surplus (the shaded area in the graph. 400

12 Application (continued) Use formula 20 with a = 10,000, b = –25, c = 500, and d = –1:

13 Summary ■ There are tables in the appendix that contain formulas to assist us in integration. ■ Sometimes we need to make substitutions before we use these tables. ■ Sometimes these formulas need to be applied repeatedly in order to complete an integration. ■ Along with previously learned methods of integration we now have a much better repertoire for integrating more complicated functions.