Antiderivatives

Think About It

Suppose this isthe graph of thederivative of a function

What do we know aboutthe original function?• Critical numbers• Where it is increasing, decreasing

What do we not know?

2

f '(x)

The work to this point has involved finding and

applying the first or second derivative of a function. In this chapter we will reverse the process. If we know the derivative of a function how

do we obtain the original function? The process is

called antidifferentiation or integration.

Anti-DerivativesDerivatives 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 = f(x) • So F(x) = 12x2 is the antiderivative of f(x) = 24x

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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

5

4 412( ) 3

4F x x x

4 412( ) 3

4F x x x

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 foreach 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 6

This indicates that multiple

antiderivatives could exist from one derivative

This indicates that multiple

antiderivatives could exist from one derivative

Indefinite Integral

The family of antiderivatives of a function f indicated by

The symbol is a stylized S to indicate summation

7

( )f x dx

Indefinite Integral

The indefinite integral is a family of functions

The + C represents an arbitrary constant• The constant of integration

8

3 41

4x dx x C

2 13 4 3 4x dx x x C

Properties of Indefinite Integrals

The power rule

The integral of a sum (difference) is the sum (difference) of the integrals

9

( ) ( ) ( ) ( )f x g x dx f x dx g x dx

11, 1

1n nx dx x C n

n

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 xdx

( ) ( )f x dx xk k f x d

• Example : Evaluate (5x 4x3 )dx.

(5x 4x3 )dx 5x dx 4x3dx 5 x dx 4 x3dx 5

x2

2 4

x4

4C

5

2x2 x4 C

• Example : Find the function f such that

• First find f (x) by integrating.

f (x) x2 and f ( 1) 2.

f (x) x2dxf (x)

x3

3C

• Example : Evaluate and check by differentiation:

2 4

3 11 dxx x

Examples

Determine the indefinite integrals as specified below

14

5x dx 4 2x x dx 7 dx12x dx

Integrate

dttt 13

dxx33 dxxx 872 2

Find each antiderivative

dx

xxx

23

5

4

3 25

dxx

1 dxx2

1

dx

xx 43

54

dxx

Find each antiderivative

dyyy 43 42

dxxx

321 dxx dxx 232

dx

x

xx4

2 13

Find each antiderivative

dyx

xx )23)(52( dxxsin4

dxx

3

cos2 dxx2cos

5 dxxx )sin9cos4(

Find each antiderivative

dxx

x2cos

sin d22 csc2

Solve the differential equation

3)2(,23)(' fxxf

Solve the differential equation

1)3(,22)(' 2 fxxxf

Solve the differential equation

2)1(,1)4(',2)('' ffxf

Solve the differential equation

1)2(,30)5(',2)('' ffxxf

Given that the graph of f(x) passes through the point (1,6)

and that the slope of its tangent line at (x.f(x) is 2x+1, find f(6)