422 Integration (Chapter 15)

Earlier, we showed that theantiderivativeofx^2 is^13 x^3 , and that any function of the form^13 x^3 +c where

cis a constant, has derivativex^2.

We say that theindefinite integralorintegralofx^2 is^13 x^3 +c, and write

Z

x^2 dx=^13 x^3 +c.

We read this as “the integral ofx^2 with respect toxis^13 x^3 +c, wherecis a constant”.

If F^0 (x)=f(x) then

Z

f(x)dx=F(x)+c.

This process is known asindefinite integration. It is indefinite because it is not being applied to a particular

interval.

DISCOVERING INTEGRALS

Since integration is the reverse process of differentiation we can sometimes discover integrals by

differentiation. For example:

² if F(x)=x^4 then F^0 (x)=4x^3

)

Z

4 x^3 dx=x^4 +c

² if F(x)=

p

x=x

1

(^2) then F^0 (x)=^1
2 x
¡^12

1
2
p
x
)
Z
1
2
p
x
dx=
p
x+c
The following rules may prove useful:
² Any constant may be written in front of the integral sign.
Z
kf(x)dx=k
Z
f(x)dx, kis a constant
Proof: Consider differentiatingkF(x) where F^0 (x)=f(x).
d
dx
(kF(x)) =kF^0 (x)=kf(x)
)
Z
kf(x)dx=kF(x)
=k
Z
f(x)dx
² The integral of a sum is the sum of the separate integrals. This rule enables us to integrate term by
term.
Z
[f(x)+g(x)]dx=
Z
f(x)dx+
Z
g(x)dx

D Integration

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\422CamAdd_15.cdr Monday, 7 April 2014 3:58:42 PM BRIAN