toán học mathêmtics

8/12/2019 ton hc mathmtics

1/48

Chng IIChng II PhPhpp TTnhnh TTch Phnch Phn

BBii 1515 -- TTch Phn Suy Rch Phn Suy Rngng

CCcc ttch phn sau ych phn sau y ggii l tl tch phn suy rch phn suy rngng

( tch phn c cn l v hn )++++

(( loloii 11 ))

Ng Thu Lng- n tp Cao Hc

a

( )a

f x dx (2)

( ) (3)f x dx

+

(a l hng s)


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Gi trcGi trcaa ttch phn suy rch phn suy rng :ng :

( )

a

f x dx+

= b

a

dxxf )(+b

lim


Nu gii hn khi ly lim l sAA hhuu hhnn, tch phngi l hhii tt, gi trca n l AA

Nu gii hn khng tkhng tnn ttii hoc bng vv hhnn,tch phn gi l phnphn kk


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V dV d11 :: Xt tch phn suy rng 21

1dx

x

+

21

1 1

1

b bdx

xx

=

11

b

= +


2 21 1

1 1lim

b

bdx dx

x x

+

+=

1lim 1

b b

= +

1= = A

Vy ta c tch phn 21

1dx

x

+ hhii tt

v gi trca n l 11


5/48



V dV d22:: Xt tch phn suy rng

1

1dx

x

+

1

1ln

1

b bdx x

x= ln b=


1

1lim ln

bdx b

x

+

+= =

Vy ta c tch phn1

1dx

x

+ phnphn kk


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HaiHai bbii toton n i vi vii ttch phn suy rch phn suy rng :ng :

Tnh gi trtch phn



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tnh gi trtch phn suy rng ta c thdngcng thc NewtonNewton--LeibnitzLeibnitz

+b

*)*) TTnhnh gi trtgi trtch phnch phn


( ) ( )a

f x dx F x a=

( ) ( )F F a= +

lim ( ) ( )x

F x F a+

=


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hoc cng thc ttch phn tch phn tng phng phnn

axvxuxdvxu

a

+=

+)()()()(

+

a

xduxv )()(

bb b


( ) ( ) lim ( ) ( )x

u x v x u x v x+

+=

hoc cng thc i bii binn ththch hch hpp


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V dV d33 :: 21

1

1

dx

x

+

+

dng cng thc NewtonNewton--LeibnitzLeibnitz


21

1 arctan11

dx xx

+ +=+

arctan( ) arctan(1)= +

2 4 4

= =


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V dV d44 :: +

+

0 3

1 x

dx

3 2

1 A Bx C+= +

2

/ 33 /21 3/ x +=

+


2

3

1ln | 1 |

1 1 2 1ln | 1 | arctan

6 3 31 3

dx xx x C

xx

= + + +

++

30 1

dx

x

+

+

21 1 2 1ln | 1| arctan

1l

6 3 3n | |

30

1x x

x x+

= + + +

( )( )( )( )


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V dV d44 :: +

+

0 3

1 x

dx

3 2

1 A Bx C+= +

2

/ 33 /21 3/ x +=

+


2

3

1ln | 1 |

1 1 2 1ln | 1 | arctan

6 3 31 3

dx xx x C

xx

= + + +

++

( )2

2

1 2 1ln ar

11

6ctan

3 31

xC

x x

x = + + +

+


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( )2

3 20

11 1 2 1ln arctan

6 3 31 10 0

xdx x

x x x

+ + + +

= + + +

1 1arctan arctan

= +


2

3 3

=


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V dV d55 :: +

0

dxe x

0

xe

+=


0( )e e=

(0 1) 1= =


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+

0

dxxe x

V dV d66 ::

0

( )x

x d e

+

=

+


0( ) ( )0

x xe e dx x

=

0

xe dx

+

= 51( )vidu=


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6

0

xx e dx+

BBi ti tp :p :

+

2


0xex


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V dV d77 :: +

0

dxe x

2tx= dttdx 2=

0 , 0x t= =

+=+= txx t=


0 02x te edx tdt =

2=02

t

t e dt

+

=

3

0

xe dx

+

BBi ti tpp


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+

=

22

u

du

2ln 2

du

u

+=

1 1

ln 2 ln 2u

+= =

1

2=


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2 2

1x t= +2

1x t= +

22 1

dx

x x

+

2

1x t =



CCch thch th11


21

dx dt t

=+

2 1x t

x t

= =

= + = +


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CCch thch th22



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1t

x=

1x=

( 0)t



CCch thch th33


2

1dx dt

t

=

2

2

2 2

2

11 1

1 1

xt

t t

tt

= =

= =


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TTch phn suy rch phn suy rng cng cbbnn

Kho st shi tca tch phn

1( 0)dx a

+

>


a xhhii tt 1 >

phnphn kk1


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Kho st shi tca tch phn

Thng ngi ta sdng 33 tiu chutiu chunnkt hp vitch phn suy rng cbn nh gi shi tca



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BBii 1515 -- TTch Phn Suy Rch Phn Suy RngngTiu chuTiu chunn 11 : (: ( SoSo ssnhnh 11))

Xt hai tch phn suy rng

)()(0 xgxf

+

adxxf )(

+

adxxg )(


Nu hhii tt th+

a

dxxg )( +

a

dxxf )( hhii tt

Nu phnphn kk th +

a

dxxg )(+

a

dxxf )( phnphn kk


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BBii 1515 -- TTch Phn Suy Rch Phn Suy RngngTiu chuTiu chunn 22 : (: ( SoSo ssnhnh 22))


+

adxxf )(

+

adxxg )(

( ) 0 , ( ) 0f x g x


( )lim( )x

f x Kg x+

= (0 )K< < +

+

adxxg )(

+

adxxf )( ccng hng hii tthoc

ccng phnng phn kk

Nu

th


25/48



Tiu chuTiu chunn 33 : (: ( TuyTuyt t i )i )


+

adxxf )( | ( ) |

adx xf

+

+ +


u

a

x xa

xx


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( )

a

f x dx+

2) hi t ( )a

f x dx+

hi t


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( ) ( )

a b

f x dx f x dx+ +

3)

cng hi thoc cng phn k


( ) ( ) ( )b

a b a

f x dx f x dx f x dx+ +

=


28/48



V dV d88 :: Kho st shi tca tch phn

31

1

1dx

x

++


3 30 ( ) ( )1

f x g xx x

= =+

31 1

1( )g x dx dx

x

+ +

= hhii tt ( 3 1)= >

Tiu chun 1:(So snh 1)3

1

1

1dx

x

+

+

hhii tt


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30/48



5/ 22 2

1

( )g x dx dxx

+ +

= hhii tt ( 5 / 2 1)= >


Tiu chun 2:(So snh 2) 32 1

xdx

x hhii tt


31/48

+

++1 2 1xx

dxx

+

++

+

123

1

2

xx

dxx


++

+

1 3 24

11

xxdxx

+

+13

1x

dxxarctg


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)



33/48


)/1( 2x


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20032003





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Ch IICh II PhPh TT hh TT h Phh Ph


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20062006


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42/48

3 21 1m

dx

x x

+

+ 20082008




Vi gi trno ca m th tchphn hi t

Tnh gi trtch phn vi m=7/3


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11

11x

xx e dxx

+

+ Tnh :




20201010


45/48



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BBi ti tpp 11:: 21 1

dxx x

+

+ +

Tnh cc tch phn suy rng sau


BBi ti tpp 22:: 20

xx e dx

+

+

2 2ln

1

dxxxBBi ti tpp 33 ::



47/48



+

++

dx

xx 52

1

2

BBi ti tpp 44 ::

Kho st shi tca cc tch phn sau :


BBi ti tpp 55 :: +

++12

1xx

dx


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