lesson 1 antiderivatives.pptx

8/14/2019 Lesson 1 AntiDerivatives.pptx

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MATH22

Calculus2

ANTIDERIVATIVES


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OBJECTIVES:

At the end of the lesson the students are

expected to:know the relationship between differentiation

and integration;

identify and explain the different parts of the

integral operation; and

perform basic integration by applying the power

formula and the properties of the indefinite

integrals.


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A function Fis called an antiderivative (or integral)of the

functionfon a given open interval if F(x) = f(x) for everyvalue of xin the interval.

DEFINITION: ANTIDERIVATIVE (INTEGRAL)

For example, the function is an antiderivative

of on interval because for eachxin

this interval .

3

3

1)( xxF

2)( xxf ),(

)(3

1)('

23xfxx

dx

dxF

However, is not the only antiderivative

off on this interval. If we add any constant C to

, then the function

3

3

1)( xxF

3

3

1x

)(0

3

1)(' 23 xfxCx

dx

dxG


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In general, once any single antiderivative is known, the other

antiderivatives can be obtained by adding constants to the

known derivative. Thus,

are all antiderivatives of .

23

1,5

3

1,2

3

1,

3

1 3333 xxxx

2)( xxf

Theorem If F(x) is any antiderivative off(x) on an open interval,

then for any constant C the functionF(x)+Cis also anantiderivative on that interval. Moreover, each antiderivative

off(x)on the interval can be expressed in the form F(x)+C by

choosing the constant C appropriately.


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DEFINITION: THE INDEFINITE INTEGRAL

The process of finding antiderivatives is called

antidifferentiation or integration. Thus, if

then integrating (orantidifferentiating) the function

f(x) produces an antiderivative of the form F(x)+C.To

emphasize this process, we use the following integral

notation

)()( xfxFdx

d

CxFdxxf )()(


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where: The expression is called an indefinite

integral.

is called an integral sign

the function is called the integrand

and the constant C is called the constant of

integration

dxxf )(

)(xf

dx indicates thatxis the variable of integration.


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Some of the properties of the indefinite integral and basic

integration formulas, which need no proof from the fact

that these properties are also known properties of

differentiation are listed below.Properties of Indefinite Integral and Basic Integration

Formula:

1;

1

.

)(...)()()](....)()([.

)()()(.

.

1

32121

nC

n

xdxxiv

dxxfdxxfdxxfdxxfxfxfiii

CxcFdxxfcdxxcfii

Cxdxi

nn

n


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EXAMPLE

dy

y

yy

dxbxa

dxxx

dxx

3

2

3

2322

3

2

1.4

2.3

763.2

.1

2

Evaluate the following integral.


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dmmm

dxxxx

dxx

xxx

dxxx

dzz

z

e

66

3 23

5

3

3

2.10

2754368.9

47.8

2.7

1

1

.6

EXERCISES

dtat

dzzz

z

t

dt

dyyy

dxxx

3

3

4

3

3 2

5

1

3

2

23

5.5

47.4

.3

44.2

325.1

Evaluate the following integral.

lesson 1 antiderivatives.pptx

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