INDEFINITE INTEGRALS Indefinite Integral ) ( ) ( ' means ) ( ) ( x f x F dx x f x F Note1: is traditionally used for an antiderivative of ) ( dx x f f is called an indefinite integral Note2: ) ( dx x f Example : C x dx x 3 2 3 1 2 3 3 1 x C x dx d
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INDEFINITE INTEGRALS
Indefinite Integral
)()(' means )()( xfxFdxxfxF Note1: is traditionally used for
an antiderivative of )( dxxf f
is called an indefinite integralNote2: )( dxxf
Example:
Cxdxx 32
3
1 23
3
1xCx
dx
d
INDEFINITE INTEGRALS
Indefinite Integral
)( dxxf
Cxdxx 32
3
1
b
adxxf )(
3
11
0
2 dxx
definite Integral
numberfunction
function offamily or
The connection between them
ba
b
adxxfdxxf
)()(
INDEFINITE INTEGRALS
Table Indefinite Integrals
INDEFINITE INTEGRALS
TERM-092
INDEFINITE INTEGRALS
TERM-092
INDEFINITE INTEGRALS
TERM-082
INDEFINITE INTEGRALS
THE SUBSTITUTION RULE
Example
dxe x20
Example
dxe x2
cea
dxe axax 1
THE SUBSTITUTION RULE
Example
dxx)5sin(
caxa
dxax )cos(1
)sin(
THE SUBSTITUTION RULE
Find
dxxx )2cos( 43
Find
dxxx 12 2
Find
dxx 12
THE SUBSTITUTION RULE
Find
dxxx 12 2
THE SUBSTITUTION RULE
Find
dxxx 125
Find
xdxtan
definite Integral
4
012 dxx
Example
3
1 2)53( x
dx
Example
e
x
xdx1
ln
THE SUBSTITUTION RULE
Example
)1ln(
0 1
e
x
x
dxe
e
Note: dxe x 1
1
dxex 1
1
INDEFINITE INTEGRALS
TERM-112
THE SUBSTITUTION RULE
T-102
092
THE SUBSTITUTION RULE
082
THE SUBSTITUTION RULE
TERM-092
THE SUBSTITUTION RULE
The Integrals of sin2 x and cos2 x
Remark
2
2cos1sin 2 x
x
2
2cos1cos2 x
x
THE SUBSTITUTION RULE
Find
xdxsec
Find
xdxtan
T-102
THE SUBSTITUTION RULE
082
THE SUBSTITUTION RULE
082
THE SUBSTITUTION RULE
082
THE SUBSTITUTION RULE
082
THE SUBSTITUTION RULE
THE SUBSTITUTION RULE
T-102
TERM-092
THE SUBSTITUTION RULE
The Integrals of sin2 x and cos2 x
092
THE SUBSTITUTION RULE
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