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