Bai 15 - Tich Phan Suy Rong

Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng

CCc c ttch phn sau y ch phn sau y ggi i l tl tch phn suy rch phn suy rngng

( tch phn c cn l v hn )

( )f x dx (1)++++

( ( loloi i 1 1 ) )

Ng Thu Lng- n tp Cao Hc

( )a

f x dx (1)

( )a

f x dx (2)

( ) (3)f x dx

+

(a l hng s)


Gi tr cGi tr ca a ttch phn suy rch phn suy rng : ng :

( )a

f x dx+

= b

a

dxxf )(+b

lim


Nu gii hn khi ly lim l s AA hhu u hhnn , tch phn gi l hhi i tt , gi tr ca n l AANu gii hn khng tkhng tn n ttii hoc bng v v hhnn , tch phn gi l phn phn kk


V d V d 1 1 :: Xt tch phn suy rng 21

1 dxx

+

21

1 11

b bdx

xx

=

1 1b

= +


2 21 1

1 1limb

bdx dx

x x

+

+=

1lim 1b b

= +

1= = A

Vy ta c tch phn 21

1 dxx

+ hhi i tt

v gi tr ca n l 11


V d V d 22:: Xt tch phn suy rng1

1 dxx

+

1

1 ln1

b bdx x

x= ln b=


1

1 lim lnb

dx bx

+

+= =

Vy ta c tch phn 1

1 dxx

+ phn phn k k


Hai Hai bbi i toton n i vi vi i ttch phn suy rch phn suy rng :ng :

Tnh gi tr tch phn

Kho st s hi t ca tch phn




tnh gi tr tch phn suy rng ta c th dng cng thc NewtonNewton--LeibnitzLeibnitz

+ +=

b b

*) *) TTnh nh gi tr tgi tr tch phnch phn


( ) ( )a

f x dx F xa

+ +=

( ) ( )F F a= + lim ( ) ( )

xF x F a

+=

b b


hoc cng thc ttch phn tch phn tng phng phnn

axvxuxdvxu

a

+=

+)()()()(

+

a

xduxv )()(b b b


( ) ( ) lim ( ) ( )x

u x v x u x v x+

+=

hoc cng thc i bii bin n ththch hch hpp


V d V d 3 3 :: 21

11

dxx

+

+

dng cng thc NewtonNewton--LeibnitzLeibnitz+ +


21

1arctan

11dx x

x

+ +=

+arctan( ) arctan(1)= +

2 4 4pi pi pi

= =


V d V d 4 4 :: +

+

0 31 xdx

3 21

11 1xx x xA Bx C+

= +++ + 2

/ 33 /1 1

21 3/ xx x x

+=

++

+


3 211 1xx x x++ +2

31 ln | 1| 1 1 2 1ln | 1| arctan

6 3 31 3dx x

x x Cx

x

= + + + +

+

21 1x x x+ +

30 1

dxx

+

+

21 1 2 1ln | 1| arctan1l6 3 3

n | |3

01x xx x

+

= + + +

( )( )( )( )


V d V d 4 4 :: +

+

0 31 xdx

3 21

11 1xx x xA Bx C+

= +++ + 2

/ 33 /1 1

21 3/ xx x x

+=

++

+


3 211 1xx x x++ +2

31 ln | 1| 1 1 2 1ln | 1| arctan

6 3 31 3dx x

x x Cx

x

= + + + +

+

( )22

1 2 1ln ar116

ctan3 31

x Cx x

x = + +

+

+

21 1x x x+ +


( )23 20

11 1 2 1ln arctan6 3 31 1 0 0

xdx xx x x

++ +

+ = + + +

1 1arctan arctan

3 3

= +


23 3

pi=

arctan arctan3 3

= +


V d V d 5 5 :: +

0dxe x

0xe

+=


0( )e e= (0 1) 1= =


+

0dxxe xV d V d 6 6 ::

0( )xx d e

+=

x x+

+=


0( ) ( )

0x xe e dx x

+

+=

0

xe dx+

= 51 ( )vidu=


6

0

xx e dx+

BBi ti tp :p :

+

7 2 dxex x


0

7 2 dxex x


V d V d 7 7 :: +

0dxe x

2tx = dttdx 2=0 , 0x t= =

+=+= tx

x t + +

x t=


0 02x te edx tdt

+ +=

2=0

2 tt e dt+

=

3

0

xe dx+

BBi ti tpp


+


+

=

22u

du

2ln 2

duu

+=

1 1ln 2 ln 2u+

= =

12

=

2 21x t= +21x t= +

tdx dt=

22 1

dx

x x

+

2 1x t =


CCch thch th 11


21

tdx dtt

=

+

2 1x tx t

= =

= + = +

CCch thch th 22


1t

x=

1x

t=

( 0)t


CCch thch th 33


t

21dx dt

t

=

22

2 2

2

11 1

1 1

xt

t t

tt

= =

= =


TTch phn suy rch phn suy rng c ng c bbnnKho st s hi t ca tch phn

1 ( 0)a

dx ax

+>


a x

hhi i tt 1 >

phn phn k k 1



Thng ngi ta s dng 3 3 tiu chutiu chunn kt hp vi tch phn suy rng c bn nh gi s hi t ca mt tch phn cho trc


mt tch phn cho trc


Tiu chuTiu chun n 11 : ( : ( So So ssnh nh 11))

Xt hai tch phn suy rng

)()(0 xgxf +

a

dxxf )( +

a

dxxg )(


Nu hhi i t t th+

a

dxxg )( +

a

dxxf )( hhi i tt

Nu phn phn kk th +

a

dxxg )(+

a

dxxf )( phn phn kk


Tiu chuTiu chun n 22 : ( : ( So So ssnh nh 22))Xt hai tch phn suy rng

+

a

dxxf )( +

a

dxxg )(( ) 0 , ( ) 0f x g x


( )lim ( )xf x Kg x+

= (0 )K< < +

+

a

dxxg )( +

a

dxxf )( ccng hng hi i tt hoc ccng phn ng phn kk

Nu

th


Tiu chuTiu chun n 33 : ( : ( TuyTuyt t i )i )Xt hai tch phn suy rng

+

a

dxxf )( | ( ) |a

dx xf+

Nu hhi i tt| ( ) | dx xf+

+dxxf )( hhi i ttth


Nu hhi i tt| ( ) |a

dx xf a

dxxf )( hhi i ttth



( )a

f x dx+

2) hi t ( )a

f x dx+

hi t


( ) ( )a b

f x dx f x dx+ +

3)

cng hi t hoc cng phn k


( ) ( ) ( )b

a b af x dx f x dx f x dx

+ +

=


V d V d 8 8 : : Kho st s hi t ca tch phn

31

11

dxx

+

+ta c 1 1


ta c 3 31 10 ( ) ( )

1f x g x

x x = =

+

31 1

1( )g x dx dxx

+ += hhi i tt ( 3 1) = >

Tiu chun 1:(So snh 1) 31

11

dxx

+

+hhi i tt


V d V d 9 9 : : Kho st s hi t ca tch phn 32 1

x dxx

+

3x

310 ( ) ; 0 ( )

1xf x g x

x x = =

5/21

x=


3( ) 1lim lim 1( )x xf x xg x

x+ +

=

5/2

3.lim

1xx x

x+=

1 K= =3

5/2

1lim 1x

x

x

x

+

=

3.lim

1xx x Kx

+= =

( )lim ( )xf xg x+


5/22 2

1( )g x dx dxx

+ += hhi i tt ( 5 / 2 1) = >

x+


Tiu chun 2:(So snh 2) 32 1

x dxx

+

hhi i tt

+

++12 1xx

dxx

+

++

+

123 12

xx

dxx

+ +2 1dxx


+

++

+

1 3 24

2

1

1

xx

dxx

+

+13 1x

dxxarctg

)Ng Thu Lng- n tp Cao Hc


)/1( 2x

+

1

cos dxx

x

Bi tp :Bi tp :

+ ln dxx 1)(xg =


2 2

dxx 2/3

)(x

xg =

20032003



20042004



20052005



1



20062006



20072007

3 21 1mdx

x x

+

+ 20082008



Vi gi tr no ca m th tch phn hi t

Tnh gi tr tch phn vi m=7/3

22 ( 1) 1mdx

x x

+

+ 20020099



Vi gi tr no ca m th tch phn hi t

Tnh gi tr tch phn vi m=1

11 11x

xx e dxx

+

+

Tnh :



20201010

20112011)) Chng minh tch phn suy rng hi t v tnh tch phn

2 2( 1)( 3 1)x x x dx+ +



6 31 4 1

dxx x+


BBi ti tp p 11:: 21 1

dxx x

+

+ +

+

Tnh cc tch phn suy rng sau


BBi ti tp p 22:: 20

xx e dx+

+

2 2ln1 dx

xxBBi ti tp p 3 3 ::


+

++dx

xx 521

2BBi ti tp p 4 4 ::

Kho st s hi t ca cc tch phn sau :


BBi ti tp p 5 5 :: +

++12 1xx

dx


BBi ti tp p 6 6 ::1 1

dxx

+

+

BBi ti tp p 7 7 :: 4 2( 1)x dx+ +

+ +


BBi ti tp p 7 7 :: 4 21 1x x

+ +

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Bai 15 - Tich Phan Suy Rong