Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

18.6 Calculus 623

Example 18.16 Evaluate

We use rule 5, a#f(x)dxaf(x)dx, from Table 18.7 to solve this problem. For the

given problem,a5 andf(x)  3 x

2

 20 x, then using the results of Example 18.15, we get

5(3x

2

 20 x)dx5(x

3

 10 x

2

C).

Example 18.17 Evaluate

We use rule 6, [f(x)g(x)]dxf(x)dxg(x)dx, from Table 18.7 to solve this

problem, as shown.

 1 x

3

 10 x

2

C 12  1 x

5

 5 xC 22

(^)  313 x
2
 20 x 2  15 x
4
 524 dx 13 x
2
 20 x 2 dx 15 x
4
 52 dx
^313 x
2
 20 x 2  15 x
4
 524 dx.
^513 x
2
 20 x 2 dx.
TABLE 18.7 Summary of Basic Integral Rules
Definitions and Rules Explanation
1 The integral of a constanta.
2 True forn 1 (see Example 18.15).
3 True forx 0 (see Example 18.18).
4 The rule for exponential function.
5 Whenaconstant (see Example 18.16).
6 See Example 18.17.
7 The substitution method.
8 The substitution method (see Example 18.20).
(^9)  The substitution method.
u¿ 1 x 2
u 1 x 2
dxln 0 u 1 x 20 C

e

u 1 x 2

u

œ

1 x 2 dxe

u 1 x 2

C

^3 u^1 x^24

n

u

œ

1 x 2 dx

3 u 1 x 24

n 1

n 1

C

^3 f^1 x^2 g^1 x^24 dxf^1 x^2 dxg^1 x^2 dx

a

#f 1 x 2 dxa

f^1 x^2 dx

e

ax

dx

1

a

e

ax

C



a

x

dxa ln 0 x 0 C

x^

n

dx

1

n 1

x

n 1

C

a^ dxaxC

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀