4.1 2012 Pearson Education, Inc. All rights reserved Slide 4.1-1 Antidifferentiation OBJECTIVE Find an antiderivative of a function. Evaluate indefinite

4.1

2012 Pearson Education, Inc.All rights reserved Slide 4.1-1

Antidifferentiation

OBJECTIVE• Find an antiderivative of a function.• Evaluate indefinite integrals using the basic integration formulas.

• Use initial conditions, or boundary conditions, to determine an antiderivative.

2012 Pearson Education, Inc. All rights reserved Slide 4.1-2

THEOREM 1The antiderivative of is the set of functions

such that

The constant C is called the constant of integration.

4.1 Antidifferentiation

f x F x C

.dF x C f x

dx


Integrals and IntegrationAntidifferentiating is often called integration.To indicate the antiderivative of x2 is x3/3 +C, we

x2 dx x3

3C,write where the notation f x dx

is used to represent the antiderivative of f (x).

f x dx F x C,More generally, where

F(x) + C is the general form of the antiderivative of f (x).



Example 1: Determine these indefinite integrals. That is, find the antiderivative of each integrand:

a.)

b.)

c.)

d.)


8dx23x dx

xe dx1

dxx

8x C Check: 8 8d

x Cdx

3x C 3 2Check: 3d

x C xdx

xe C Check: x xde C e

dx

ln x C 1Check: ln

dx C

dx x


1. k dx kx C (k is a constant)

2. xrdx xr1

r 1C, provided r 1

(To integrate a power of x other than 1, increase the

power by 1 and divide by the increased power.)

THEOREM 2: Basic Integration Formulas



3. x 1dx 1

xdx

dx

x ln x C, x 0

x 1dx ln x C, x 0

(We will generally assume that x 0.)

4. beaxdx b

aeax C

THEOREM 2: Basic Integration Formulas (continued)



Example 2: Use the Power Rule of Antidifferentiation to determine these indefinite integrals:

a.)

b.)


7 993

1a) ; b) ; c) ; d) x dx x dx xdx dx

x 7 1

7 81

7 1 8

xx dx C x C

8 7 71 1Check: 8

8 8

dx C x x

dx

99 199 1001

99 1 100

xx dx C x C

100 99 991 1Check: 100

100 100

dx C x x

dx


Example 2 (Continued)

c) We note that Therefore


1/2 11/2 3/22

1 312

xxdx x C x C

3/2 1/2 1/22 2 3Check:

3 3 2

dx C x x x

dx

1/2.x x


Example 2 (Concluded)

d) We note that Therefore


3 13 2

3 2

1 1 1

3 1 2 2

xdx x dx C x C C

x x

2 3 33

1 1 1Check: 2

2 2

dx C x x

dx x

33

1.x

x


4.1 AntidifferentiationQuick Check 1

Determine these indefinite integrals:

a.)

b.)

c.)

d.)

10x dx

200x dx

6 x dx

4

1dx

x

10 1

10 1

xC

111

11x C

200 1

200 1

xC

2011

201x C

1/6x dx1/6 1

11

6

xC

7/66

7x C 6 76

7x C

4x dx4 1

4 1

xC

31

3x C

3

1

3C

x


Example 3: Determine the indefinite integral

Since we know that it is reasonable to make this initial guess:

But this is (slightly) wrong, since


4 .xe dx

,x xde e

dx

4 4 .x xe dx e C

4 44x xde C e

dx


Example 3(Concluded): We modify our guess by

inserting to obtain the correct antiderivative:

This checks:


1

44 41

4x xe dx e C

4 4 41 14

4 4x x xd

e C e edx


4.1 AntidifferentiationQuick Check 2

Find each antiderivative:

a.)

b.)

3xe dx

1/2 xe dx

31

3xe C

1/22 xe C


THEOREM 3Properties of Antidifferentiation

(The integral of a constant times a function

is the constant times the integral of the function.)

(The integral of a sum or difference is the

sum or difference of the integrals.)

P1. ( ) ( )kf x dx k f x dx

P2. ( ) ( ) ( ) ( )f x g x dx f x dx g x dx



Example 4: Determine these indefinite integrals. Assume x > 0.

a.) We antidifferentiate each term separately:


4

5 2 4 3 2a) 3 7 8 ; b)

x xx x dx dx

x

5 2 5 23 7 8 3 7 8 x x dx x dx x dx dx

6 31 7 8

2 3 x x x C

6 31 1 3 7 8

6 3

x x x


Example 4 (Concluded):b) We algebraically simplify the integrand by noting

that x is a common denominator and then reducing each ratio as much as possible:

Therefore,


4 434 3 2 4 3 2 4

3 2x x x x

xx x x x x

434 3 2 4

3 2

x x

dx x dxx x

4

4

1 4ln 3 2

41

4ln 32

x x x C

x x x C



Quick Check 3

Determine these indefinite integrals:

a.)

b.)

c.)

4 3 22 3 7 5x x x x dx

25x dx

2

2

7 2

x xdx

x

5 4 3 22 3 7 15

5 4 3 2x x x x x C

2 10 25x x dx 3 215 25

3x x x C

2

7 21 dx

x x

127ln

1

xx x C

271 2x dx

x

27lnx x C

x


Example 5: Find the function f such that

First find f (x) by integrating.

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


( )f x 2x dx

( )f x 3

3

xC


Example 5 (concluded): Then, the initial condition allows us to find C.

Thus, f (x) x3

3

7

3.


( 1)f 2

3( 1)

3C

2 1

3C

C7

3



Section Summary

• The antiderivative of a function is a set of functions F(x) such

that

where the constant C is called the constant of integration.

• An antiderivative is denoted by an indefinite integral using the integral sign, If is an antiderivative of we write

We check the correctness of an antiderivative we have found by differentiating it.

f x

,dF x C f x

dx

. F x f x

.f x dx F x C



Section Summary Continued

• The Constant Rule of Antidifferentiation is

• The Power Rule of Antidifferentiation is

• The Natural Logarithm Rule of Antidifferentiation is

• The Exponential Rule (base e) of Antidifferentiation is

• An initial condition is an ordered pair that is a solution of a particular antiderivative of an integrand.

.k dx kx C

11, for 1.

1n nx dx x C n

n

1ln , for 0.dx x C x

x

1, for 0.ax axe e C a

a

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4.1 2012 Pearson Education, Inc. All rights reserved Slide 4.1-1 Antidifferentiation OBJECTIVE Find an antiderivative of a function. Evaluate indefinite