13.1 Antiderivatives and Indefinite Integrals

The AntiderivativeThe reverse operation of finding a derivative is called the

antiderivative. A function F is an antiderivative of a function f if

F ’(x) = f (x).

1) Find the antiderivative of f(x) = 5

Find several functions that have the derivative of 5

Answer: 5x; 5x+ 1; 5x -3;

2) Find the antiderivative of f(x) = x2

Find several functions that have the derivative of x2

Answer: exorxorx 333

3

1;

3

1;

3

1

Theorem 1:

If a function has more than one antiderivative, then the antiderivatives differ by at most a constant.

• The graphs of antiderivatives are vertical translations of each other.

• For example: f(x) = 2x

Find several functions that are the antiderivatives

for f(x)

Answer: x2,

x2 + 1,

x2 + 3,

x2 - 2,

x2 + c (c is any real number)

The symbol is called an integral sign, and the function f (x) is called the integrand. The symbol dx indicates that anti-differentiation is performed with respect to the variable x.By the previous theorem, if F(x) is any antiderivative of f, then

The arbitrary constant C is called the constant of integration.

Indefinite Integrals

CxFdxxf )()(

Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol dxxf )(

Indefinite Integral Formulas and Properties

dxxgdxxfdxxgxf

dxxfkdxxfk

Cxdxx

Cedxe

nCn

xdxx

xx

nn

)()()()(.5

)()(.4

||ln1

.3

.2

1,1

.11

(power rule)

It is important to note that property 4 states that a constant factor can be moved across an integral sign. A variable factor cannot be moved across an integral sign.

Example 1:

A)

B)

C)

dtet16

dxx43

dx2 Cx 2

Cet 16

CxCx

55

5

3

53

Example 1 (continue)

D)

dxdxxdxx 132 25

dxxx )132( 25

Cxxx

1

33

62

36

Cxxx 36

3

1

Example 1 (continue)

E)

dxedxx

x45

dxe

xx4

5

Cex x 4ln5

dxedxx

x41

5

Example 2

A)

dxxdxx 43

2

32

dx

xx

43

2 32

Cxx

33

35

233

5

dxxdxx 43

2

32

Cxx 33

5

5

6

Cx

x 3

3

5 1

5

6

Example 2 (continue)

B)

dww5

3

4 dww5 34

Cw

58

45

8

Cx 5

8

2

5

Example 2 (continue)

C)

xdxdxx 82

dx

x

xx2

34 8

Cxx

28

3

23

dxxx 82

Cxx

23

43

Example 2 (continue)

D)

dxxdxx 2

1

3

1

68

dxx

x6

83

Cxx

21

6

34

82

1

3

4

Cxx 126 3

4

Example 2 (continue)

E)

dxxdxdxxdxx 623 23

dxxx )3)(2( 2

Cxxxx

6

22

33

4

234

dxxxx 623 23

Cxxxx

64

234

Example 3Find the equation of the curve that passes through (2,6) if its

slope is given by dy/dx = 3x2 at any point x.

The curve that has the derivative of 3x2 is

Since we know that the curve passes through (2, 6), we can find out C

dxx23

Cxy 3

CxCx

33

33

C 326

C862C

Therefore, the equation is y = x3 - 2

Example 4Find the revenue function R(x) when the marginal revenue is

R’(x) = 400 - .4x and no revenue results at a 0 production

level. What is the revenue at a production of 1000 units?

The marginal revenue is the derivative of the function so to find the

revenue function, we need to find the antiderivative of that function

So R(x) = 400x -.2x2, we know need to find R(1000)

dxx)4.400( 22

2.4002

4.400 xxCx

x

000,200)1000(2.)1000(400)1000( 2 R

Therefore, the revenue at a production level of 1000 units is $200,000

xdxdx 4.400

Example 5The current monthly circulation of the magazine is 640,000 copies. Due

to the competition from a new magazine, the monthly circulation is

expected to decrease at a rate of C’(t)= -6000t1/3 copies per month, t is

the # of months. How long will it take the circulation of the magazine to

decrease to 460,000 copies per month?

We must solve this equation: C(t) = 460,000 with C(0) = 640,000

To find the function C(t), take the antiderivative

dtt 3

1

6000 CtCt

3

43

4

4500

34

6000

C

C

CC

000,640

04500000,640

04500)0(

3

4

3

4

000,6404500)( 3

4

ttC

t

t

t

t

t

9.15

40

40

4500000,180

000,6404500000,460

4

3

3

4

3

4

3

4

3

4

4

3

So, it takes about 16 months