082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

432 Chapter 13 Integration
(f) TheindependentvariableiszbutfromTable13.1wecandeduce ∫ e az dz = eaz
a +c.
Hence, takinga = −3 we obtain
∫
e −3z dz = e−3z
−3 +c=−e−3z 3 +c
(g) Since 1 x 2 =x−2 ,wefind
∫ 1
x 2 dx = ∫
x −2 dx = x−1
−1 +c=−1 x +c
(h) Whenintegratingcos100nπt withrespecttot,notethat100nπisaconstant.Hence,
usingpart (b)wefind
∫
cos100nπt dt = sin100nπt
100nπ +c
13.2.1 Integrationasalinearoperator
Integration, like differentiation, is a linear operator. If f and g are two functions of x,
then
∫ ∫ ∫
f+gdx= fdx+ gdx
This states that the integral of a sum of functions is the sum of the integrals of the individual
functions.IfAisaconstant and f a functionofx, then
∫
∫
Afdx=A
f dx
Thus,constantfactors can betakenthroughthe integral sign.
IfAandBareconstants,and f andgarefunctions ofx, then
∫
∫
Af+Bgdx=A
∫
fdx+B
gdx
These three properties are all consequences of the fact that integration is a linear operator.
Note that the first two are special cases of the third. The properties are used in
Example 13.4.
Example13.4 Use Table 13.1 and the properties of a linear operator to integrate the following
expressions:
(a) x 2 +9 (d) (t +2) 2 (g) 3sin4t
(b) 3t 4 − √ t
(c)
1
x
(e)
(f) 4e 2z
1
+z (h) 4cos(9x +2)
z
(i) 3e 2z