反微分與不定積分 及其性質 1. 反微分 (Antiderivatives) 2. 不定積分 (Indefinite...
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Transcript of 反微分與不定積分 及其性質 1. 反微分 (Antiderivatives) 2. 不定積分 (Indefinite...
反微分與不定積分及其性質
1. 反微分 (Antiderivatives)
2. 不定積分 (Indefinite Integral)
3. 積分規則 (Rules of Integration)
4. 替代法 (Substitution) page 358~379
Definition of Antiderivative:
If F’(x)=f(x), then F(x) is an antiderivative of f(x)
Example
EX1(a)
F(x)=10x, F’(x)=10
F(x) is an antiderivative of f(x)=10
Example
EX1(b)
F(x)= , F’(x)=2x
F(x) is an antiderivative of f(x)=2x
2x
Example
EX2 Find an antiderivative of f(x)=
,
is an antiderivative of
When n=5, is an antiderivative of
An antiderivative of is
45x
( ) ng x x 1( ) ng x nx
1( ) ng x nx ( ) ng x x
nx 45x
nx11
1nx
n
Example
EX3 Find an antiderivative of f(x)=
let F(x)= and F’(x)= =f(x)
F(x) is an Antiderivative of f(x)
xexe xe
Notice
F(x)= and F’(x)= let G(x)= +2 and G’(x)= =f(x)
let H(x)= +100 and H’(x)= =f(x)
F(x), G(x) and H(x) are antiderivative of f(x)
Antiderivatives of f(x) differ by a constant
xexexe
xe
xe
xe
xe
Property of Antiderivatives
If F(x) and G(x) are both antiderivatives of a function f(x) on an interval, then there is a constant C such that
F(x)-G(x)=C
The arbitrary real number C is called an integration constant (積分常數 )
EX: F(x)=2x+2 and F’(x)=2 G(x)=2x+100 and G’(x)=2 H(x)=2x+10000 and H’(x)=2
Indefinite Integral
The family of all antiderivative of f(x) (F’(x) = f(x)) is indicated by
where C : Integral Constant : Integral Sign(積分符號 ) f(x) : Integrand(積分函數 ) dx : integral of f(x) with respect to x
is called an Indefinite Integral
( ) ( )f x dx F x C
( )f x dx
Power Rule For any real number ,
EX4 Use the power rule to find each indefinite integral.
(a)
(b)
(c)
(d)
1n 1
1
nn xnx dx C
3 1 4
3
3 1 4
t tt dt C C
0 101
0 1
tdt t dt C t C
1 31 2 12 2
2
1 2 1 3
ttdt t dt C t C
2 12
2
1 1
2 1
tdt t dt C C
t t
Constant Multiple Rule and Sum or Difference Rule
If all indicated integrals exist,
and
for any real number a, b
( ) ( )a f x dx a f x dx
( ) ( ) ( ) ( )af x bg x dx a f x dx b g x dx
Example: Find 3/4)( xxf 4/3 7 /33
7x dx x C Example: Find
because are constants. Then we can use C to represent integral constant where
4x)dx(3x2
xdxdxxdxxx 43)43( 22
)2
(4)3
(3 2
2
1
3
Cx
Cx
Cx
x 2
23
21,CC
1 2C C C
EX5 Use the rules to find each integral.
(a) (by constant multiple rule)
(by power rule)
(b)
3 32 2d d 3 1
412
3 1 2C C
55
1212dz z dz
z 5 1
412 35 1
zC z C
EX6 Use the rules to find each integral
(c)
22 4 21 2 1x dx x x dx 4 1 2 1
24 1 2 1
x xx C
EX5 Use the rules to find each integral.
(a) (by constant multiple rule)
(by power rule)
(b)
3 32 2d d 3 1
412
3 1 2C C
55
1212dz z dz
z 5 1
412 35 1
zC z C
Review of Derivative of Exponential Function
f(x) = f’(x) =
f(x) = f’(x) =
f(x) = f’(x) =
f(x) = f’(x) =
xe xe
lnx x aa e ln xa a
kxe
lnkx kx aa e
kxke
ln kxk a a
Indefinite Integrals of Exponential Functions
,
,
kxkx e
e dx Ck
x xe dx e C 0k
0k
ln
xx a
a dx Ca
(ln )
kxkx a
a dx Ck a
Review of Derivative of Exponential Function
f(x) = f’(x) =
f(x) = f’(x) =
f(x) = f’(x) =
f(x) = f’(x) =
xe xe
lnx x aa e ln xa a
kxe
lnkx kx aa e
kxke
ln kxk a a
Indefinite Integrals of Exponential Functions
,
,
kxkx e
e dx Ck
x xe dx e C 0k
0k
ln
xx a
a dx Ca
(ln )
kxkx a
a dx Ck a
EX7 Exponential Functions
(a)
(b)
(c)
(d)
99
9
xx e
e dx C
9 9 9t t te dt e dt e C
545 5
4 4
54
123 3
5
uu ue
e du C e C
5 5
5 2 22
5 ln 2 5 ln 2
x xx dx C C
Indefinite Integrals of Exponential Functions
,
,
kxkx e
e dx Ck
x xe dx e C 0k
0k
ln
xx a
a dx Ca
(ln )
kxkx a
a dx Ck a
EX7 Exponential Functions
(a)
(b)
(c)
(d)
99
9
xx e
e dx C
9 9 9t t te dt e dt e C
545 5
4 4
54
123 3
5
uu ue
e du C e C
5 5
5 2 22
5 ln 2 5 ln 2
x xx dx C C
Indefinite Integral of
Derivative of Logarithmic Function
f(x) = where
f’(x) =
ln x 0x 11
xx
1 1 lnxx dx dx x C
1
ln , 1
, 11
n n
x nx dx x
nn
1x
EX8 Integrals
(a)
(b) 2 25 125lnx x
x e dx x e C
4 14 4lndx dx x C
x x
EX9 Cost-1
Suppose a publishing company has found that th
e marginal cost at a level of production of x thous
and books is given by
and that the fixed cost (the cost before the firsr bo
ok can be produced) is $25,000. Find the cost fu
nction 2 25 125lnx x
x e dx x e C
50'( )C x
x
EX9 Cost-2
and use the indefinite integral rules to integrate the
function
When x=0, C(0)=25,000, K=25,000
The cost function is
121 1
2 2
1
12
50 50 1001
xx dx K x K
12
50'( ) 50C x x
x
( ) 100 25,000C x x
Review of the Chain Rule
1. Let u = g(x), =g’(x)
2. Let w = f(g(x))=f(u), =f’(u)
3.
f(g(x)) = f’(u) g’(x)=f’(g(x))g’(x)
df g x f g x g x
dx
dw dw dudx du dx
dudx
dwdu
Substitution Rules
If u=g(x) is a differential function where du=g’(x)dx, then
EX:
1. , du=2xdx
2.
'( ( )) '( ) ( ( ))F g x g x dx F g x C 52 4 210 ( 1) 1x x dx x C
2 1u x 2 4 410 ( 1) 5x x dx u du
5 2 5( 1)u C x C
General Power Rule for Integrals
For u=f(x) and du=f’(x)dx,
EX1: Find
let u = , du =6x
1
1
nn u
u dx Cn
2 46 (3 4)x x dx23 4x
2 4 2 46 (3 4) (3 4) (6 )x x dx x xdx 5
4 2 51(3 4)
5 5
uu du C x C
EX2 General Power Rule
Find
Let u= , 121
2
13 2
12
1( )1
ux x dx u du C
2 3 1x x dx3 1x 23du x dx
3322 32 2
13 3
u C x C
EX3 General Power Rule
Find
Let u= ,
22
3
6
xdx
x x
22
3
6
xdx
x x
2 6x x 2 6du x dx
1216
2x x C
22 2
1 2( 3) 1
2 ( 6 ) 2
xdx u du
x x
11
2 1
uC
Indefinite Integrals of
For u=f(x) and du=f’(x)dx,
Indefinite Integrals of For u=f(x) and du=f’(x)dx,
u ue dx e C
ue
1u
1 lndu
u dx u Cu
Example of Substitution
EX4: Find
let u = , du =
32 xx e dx3x
23x dx
3 32 21 1(3 )
3 3x x ux e dx e x dx e du
31 1
3 3u xe C e C
Example of Substitution
EX5: Find
let u = ,
2
12 3
3x dx
x x
2
2 3
3
xdx
x x
2 3x x 2 3du x dx
1 2ln ln 3u du u C x x C
EX6 Substitution
Find
Let u = , , x=1-u
31 12 2 21 (1 )x xdx u u du u u du
1x xdx1 x du dx
312 2 3 5
2 2
1 1
312 2
2 2
1 1 3 5
u uC u u C
EX Integrals
Find
Let
4 30 3( 3 ) (4 3)x x x dx xxxg 3)( 4 34)( 3 xxg
dxxgxgdxxxx )()()34()3( 303304
Cxg 31)]([ 31
Cxx 31
)3( 314
Example: Find
dxxxx )126()6( 253
Let xxu 63
dudxxdxx
dxxdu
2)63(2)126(
)63(22
2
duudxxxx 2)126()6( 5253
C
Cxx
u
6
)6(
6
63
6
2
2
EX Integrals
Find
Let xxg cos)( xdxx cossin10
g(x)=sin x
dxxgxgxdxx )()(cossin 1010
Cxg
11
)(11
Cx
11
sin11
Example: Find
42 xu xdxdu 2
xdxxxdxx 2)4()4( 21102102
Cduu u 112110
21 11
xdxx 102 )4(
Example: Find
42 xu xdxdu 2
xdxxxdxx 2)4()4( 21102102
Cduu u 112110
21 11
Cx 22
)4( 112
xdxx 102 )4(
Example: Find
It’s not suitable to apply the substitution rules
dxxxdxx xx 224
222 )93()3(
42
dxxx 222 )3(2
Cxxx
dxxxx
35
7
246
35
3
28
)934
(
Substitution Method
The choice of u is one of the following:
1. The quantity under a root or raised to a
power
2. The exponent on e
3. The quantity in the denominator