2. THINK ABOUT IT
Suppose this is
the graph of the
derivative of a function
What do we know about
the original function?
Critical numbers
Where it is increasing, decreasing
What do we not know?
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In this chapter we will
learn how to determine
the original function when
given the derivative.
We will also learn some
applications.
f '(x)
3. ANTI-DERIVATIVES
Derivatives give us the rate of change of a
function
What if we know the rate of change …
Can we find the original function?
If f '(x) = f(x)
Then f(x) is an antiderivative of f’(x)
Example – let f(x) = 12x2
Then f '(x) = 24x
So f(x) = 12x2 is the antiderivative of f’(x) = 24x
3
4. FINDING AN ANTIDERIVATIVE
Given f(x) = 12x3
What is the antiderivative, f(x)?
Use the power rule backwards
Recall that for f(x) = xn … f '(x) = n • x n – 1
That is …
Multiply the expression by the exponent
Decrease exponent by 1
Now do opposite
(in opposite order)
Increase exponent by 1
Divide expression by new exponent
4
4 4
12
( ) 3
4
F x x x
5. FAMILY OF ANTIDERIVATIVES
Consider a family of parabolas
f(x) = x2 + n
which differ only by value of n
Note that f '(x) is the same for
each version of f
Now go the other way …
The antiderivative of 2x must be different for
each of the original functions
So when we take an antiderivative
We specify F(x) + C
Where C is an arbitrary constant
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This indicates that
multiple
antiderivatives
could exist from
one derivative
6. INDEFINITE INTEGRAL
The family of antiderivatives of a function f indicated by
The symbol is a stylized S to indicate summation
6
( )
f x dx
7. INDEFINITE INTEGRAL
The indefinite integral is a family of functions
The + C represents an arbitrary constant
The constant of integration
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3 4
1
4
x dx x C
2 1
3 4 3 4
x dx x x C
8. PROPERTIES OF INDEFINITE
INTEGRALS
The power rule
The integral of a sum (difference) is the sum (difference)
of the integrals
8
( ) ( ) ( ) ( )
f x g x dx f x dx g x dx
1
1
, 1
1
n n
x dx x C n
n
9. PROPERTIES OF INDEFINITE
INTEGRALS
The derivative of the indefinite integral is the original
function
A constant can be factored out of the integral
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( ) ( )
d
f x dx f x
dx
( ) ( )
f x dx x
k k f x d
10. TRY IT OUT
Determine the indefinite integrals as specified below
10
3 5
x dx
1/ 4
t dt
6 dx
3 2
12 6 8 5
y y y dy
