Cambridge Additional Mathematics

416 Integration (Chapter 15)

The process of findingyfrom

dy

dx

,orf(x) from f^0 (x), is the reverse process of

differentiation. We call itantidifferentiation.

y or f(x)

dy

dx

or f^0 (x)

Consider

dy

dx

=x^2.

From our work on differentiation, we know that when we differentiate power functions the index reduces

by 1. We hence know thatymust involvex^3.

Now if y=x^3 then dy

dx

=3x^2 , so if we start with y=^13 x^3 then dy

dx

=x^2.

However, for all of the cases y=^13 x^3 +2, y=^13 x^3 + 100, and y=^13 x^3 ¡ 7 , we find that

dy

dx

=x^2.

In fact, there are infinitely many functions of the form y=^13 x^3 +c wherecis an arbitrary constant, which

will give

dy

dx

=x^2. Ignoring the arbitrary constant, we say that^13 x^3 is theantiderivativeof x^2 .Itis

the simplest function which, when differentiated, gives x^2.

If F(x) is a function where F^0 (x)=f(x) we say that:

² thederivativeof F(x) is f(x) and

² theantiderivativeof f(x) is F(x).

Example 3 Self Tutor

Find the antiderivative of: a x^3 b e^2 x c

1

p

x

a d

dx

¡

x^4

¢

=4x^3

) d

dx

¡ 1

4 x

4 ¢=x 3

) the antiderivative of x^3 is^14 x^4.

b d

dx

¡

e^2 x

¢

=e^2 x£ 2

) d

dx

¡ 1

2 e

2 x¢= 1

2 £e

2 x£2=e 2 x

) the antiderivative of e^2 x is^12 e^2 x.

c

1

p

x

=x

¡^12

Now

d

dx

(x

1

(^2) )=^1
2 x
¡^12
)
d
dx
(2x
1
(^2) )=2(^1
2 )x
¡^12
=x
¡^12
) the antiderivative of
1
p
x
is 2
p
x.
differentiation
antidifferentiation
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\416CamAdd_15.cdr Monday, 7 April 2014 3:58:00 PM BRIAN