13 - The Integral.pptx

7/27/2019 13 - The Integral.pptx

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The Integral


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Suppose f(x) = x

3

+ 100, what is f (x) ? Suppose g(x) = x3 + 10, what is g(x) ?

Suppose h(x) = x3 1, what is h(x) ?

Antiderivatives

Definition

A functionFis called an antiderivative offonan interval IifF(x) =f(x) for allx inI.


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IfFis an antiderivative offon an intervalI, then

the most general antiderivative offonIis

F(x) + C

where Cis an arbitrary constant.

Theorem

Antiderivatives


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Family of Functions

By assigning specific values to C, we obtain afamily of functions.

Their graphs are vertical

translates of one another.

This makes sense, as each

curve must have the sameslope at any given value

ofx.


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Notation for Antiderivatives

The symbol is traditionally used to

represent the most general an antider ivative of f

on an open intervaland is called the indefinite

integral of f.

Thus, means F(x) =f(x)

( )f x dx

( ) ( ) F x f x dx


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( )f x dxThe expression:read the indefinite integral offwith respect to

x,means to find the set of all antiderivatives off.

( )f x dx

Integral sign Integrand

x is called the variable

of integration

Indefinite Integral


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For example, we can write

Thus, we can regard an indefinite integral as representingan entire family of functions (one antiderivative for each

value of the constant C).

3 32 2because

3 3

x d xx dx C C x

dx

Indefinite Integral


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Every antiderivativeFoffmust be of the form

F(x) = G(x) + C, where Cis a constant.

Example:26 3xdx x C

Represents every possible

antiderivative of 6x.

Constant of Integration


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1

if 11

nn xx dx C n

n

Example:

43

4

xx dx C

Power Rule for the Indefinite

Integral


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1 1 lnx dx dx x Cx

x x

e dx e C

Indefinite Integral ofexand bx

ln

xx b

b dx C b

Power Rule for the Indefinite

Integral


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Sum and Difference Rules

f g dx fdx gdx

Example:

2 2x x dx x dx xdx

3 2

3 2

x xC


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( ) ( )kf x dx k f x dx ( constant)k

4 43 32 2 2

4 2

x xx dx x dx C C

Constant Multiple Rule

Example:


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Example - Different Variable

Find the indefinite integral:

273 2 6ue u du

u

213 7 2 6ue du du u du du

u

323 7ln 63

ue u u u C


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Exercises:

dx5.3

dxxx )334(.723

dxx3)12(2.10

dx

.2

dxx5

1

.4

dx.1

dxx3)5(.5

3.8 xdx

dxx )34(.6

Find the indefinite integral of the following:

dxxx )243(.92


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Integration by Substitution

Method of integration related to chain rule. Ifuis a function ofx, then we can use the formula

/

ff dx du

du dx


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Example: Consider the integral:

9

2 33 5x x dx3 2pick +5, then 3u x du x dx

10

10

u C 9u du

10

3 5

10

xC

Sub to get Integrate Back Substitute

Integration by Substitution


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Example: Evaluate25 7x x dx


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3

ln

dx

x x

Example: Evaluate


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3

3 2

t

t

e dt

e Example: Evaluate


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Exercises:

dxx 12.1Find the indefinite integral of the following:

dxx3 43.2

dxxx3 2 9.3

dxxx62 )12(.4

dxxx1032 )1(.5

dx

x

x54

3

)21(.7

dxxx 34

2 )44(.6

dxxx 53.854

13 - The Integral.pptx

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