13 - the integral.pptx
TRANSCRIPT
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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