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

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