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.

dxxx 16

1

||ln11

abua

abua

adu

ubau

Cx

xdx

xx

|

416

416|ln

4

1

16

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 , u2 = 9x2, du = 3dx and a = 1, we could make the necessary adjustments.

dxxx 19 22

2242222222 ln2

8

1auuaauauuduauu

7

Example(continued)

1

27u2 u2 1du

1

27

1

83x 18x 2 12 9x2 12 ln 3x 9x 2 12

C

x 2 9 x2 1 dx

1

9

1

3

9 x 2 9 x2 1 3dx

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!

dxex x3

First use:

Second use:

Third use:

un e au du

un e au

a

n

aun 1 e au du

x3 e x dx

x3 e x

1

3

1x2 e x dx

x3 e x dx x3 e x 3 x2 e x 6 x e x dx

6 x e x 6e x C

x3 e x dx x3 e x 3 x2 e x

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

5( )

500

xp S x

x

x

p 20 :

p 5x

500 x

20 5x

500 x

10,000 20x 5x

x 400

11

Application(continued)

Step 2. Sketch a graph:

Step 3. Find the producers’ surplus (the shaded area in the graph.

20p

400 x

5( )

500

xp S x

x

PS [ p S(x)]dx0

x

20 5x

500 x

dx0

400

10,000 25x

500 xdx

0

400

12

Application(continued)

Use formula 20 with a = 10,000, b = –25, c = 500, and d = –1:

a bu

c du du bu

d

ad bc

d 2ln c du

PS 25x 2,500ln 500 x |0400

10,000 2,500ln 100 2,500ln 500

$5,976

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.