Download - Bai 15 - Tich Phan Suy Rong
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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)
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Gi tr cGi tr ca a ttch phn suy rch phn suy rng : ng :
( )a
f x dx+
= b
a
dxxf )(+b
lim
Ng Thu Lng- n tp Cao Hc
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
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
V d V d 1 1 :: Xt tch phn suy rng 21
1 dxx
+
21
1 11
b bdx
xx
=
1 1b
= +
Ng Thu Lng- n tp Cao Hc
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
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
V d V d 22:: Xt tch phn suy rng1
1 dxx
+
1
1 ln1
b bdx x
x= ln b=
Ng Thu Lng- n tp Cao Hc
1
1 lim lnb
dx bx
+
+= =
Vy ta c tch phn 1
1 dxx
+ phn phn k k
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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
Ng Thu Lng- n tp Cao Hc
Kho st s hi t ca tch phn
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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
Ng Thu Lng- n tp Cao Hc
( ) ( )a
f x dx F xa
+ +=
( ) ( )F F a= + lim ( ) ( )
xF x F a
+=
b b
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
hoc cng thc ttch phn tch phn tng phng phnn
axvxuxdvxu
a
+=
+)()()()(
+
a
xduxv )()(b b b
Ng Thu Lng- n tp Cao Hc
( ) ( ) lim ( ) ( )x
u x v x u x v x+
+=
hoc cng thc i bii bin n ththch hch hpp
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
V d V d 3 3 :: 21
11
dxx
+
+
dng cng thc NewtonNewton--LeibnitzLeibnitz+ +
Ng Thu Lng- n tp Cao Hc
21
1arctan
11dx x
x
+ +=
+arctan( ) arctan(1)= +
2 4 4pi pi pi
= =
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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
+=
++
+
Ng Thu Lng- n tp Cao Hc
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
+
= + + +
( )( )( )( )
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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
+=
++
+
Ng Thu Lng- n tp Cao Hc
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+ +
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
( )23 20
11 1 2 1ln arctan6 3 31 1 0 0
xdx xx x x
++ +
+ = + + +
1 1arctan arctan
3 3
= +
Ng Thu Lng- n tp Cao Hc
23 3
pi=
arctan arctan3 3
= +
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
V d V d 5 5 :: +
0dxe x
0xe
+=
Ng Thu Lng- n tp Cao Hc
0( )e e= (0 1) 1= =
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
+
0dxxe xV d V d 6 6 ::
0( )xx d e
+=
x x+
+=
Ng Thu Lng- n tp Cao Hc
0( ) ( )
0x xe e dx x
+
+=
0
xe dx+
= 51 ( )vidu=
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
6
0
xx e dx+
BBi ti tp :p :
+
7 2 dxex x
Ng Thu Lng- n tp Cao Hc
0
7 2 dxex x
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
V d V d 7 7 :: +
0dxe x
2tx = dttdx 2=0 , 0x t= =
+=+= tx
x t + +
x t=
Ng Thu Lng- n tp Cao Hc
0 02x te edx tdt
+ +=
2=0
2 tt e dt+
=
3
0
xe dx+
BBi ti tpp
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
+
Ng Thu Lng- n tp Cao Hc
+
=
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 =
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
CCch thch th 11
Ng Thu Lng- n tp Cao Hc
21
tdx dtt
=
+
2 1x tx t
= =
= + = +
-
CCch thch th 22
Ng Thu Lng- n tp Cao Hc
-
1t
x=
1x
t=
( 0)t
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
CCch thch th 33
Ng Thu Lng- n tp Cao Hc
t
21dx dt
t
=
22
2 2
2
11 1
1 1
xt
t t
tt
= =
= =
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
TTch phn suy rch phn suy rng c ng c bbnnKho st s hi t ca tch phn
1 ( 0)a
dx ax
+>
Ng Thu Lng- n tp Cao Hc
a x
hhi i tt 1 >
phn phn k k 1
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Kho st s hi t ca tch phn
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
Ng Thu Lng- n tp Cao Hc
mt tch phn cho trc
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Tiu chuTiu chun n 11 : ( : ( So So ssnh nh 11))
Xt hai tch phn suy rng
)()(0 xgxf +
a
dxxf )( +
a
dxxg )(
Ng Thu Lng- n tp Cao Hc
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
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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
Ng Thu Lng- n tp Cao Hc
( )lim ( )xf x Kg x+
= (0 )K< < +
+
a
dxxg )( +
a
dxxf )( ccng hng hi i tt hoc ccng phn ng phn kk
Nu
th
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Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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
Ng Thu Lng- n tp Cao Hc
Nu hhi i tt| ( ) |a
dx xf a
dxxf )( hhi i ttth
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
( )a
f x dx+
2) hi t ( )a
f x dx+
hi t
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
( ) ( )a b
f x dx f x dx+ +
3)
cng hi t hoc cng phn k
Ng Thu Lng- n tp Cao Hc
( ) ( ) ( )b
a b af x dx f x dx f x dx
+ +
=
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
V d V d 8 8 : : Kho st s hi t ca tch phn
31
11
dxx
+
+ta c 1 1
Ng Thu Lng- n tp Cao Hc
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
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
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=
Ng Thu Lng- n tp Cao Hc
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+
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
5/22 2
1( )g x dx dxx
+ += hhi i tt ( 5 / 2 1) = >
x+
Ng Thu Lng- n tp Cao Hc
Tiu chun 2:(So snh 2) 32 1
x dxx
+
hhi i tt
-
+
++12 1xx
dxx
+
++
+
123 12
xx
dxx
+ +2 1dxx
Ng Thu Lng- n tp Cao Hc
+
++
+
1 3 24
2
1
1
xx
dxx
+
+13 1x
dxxarctg
-
)Ng Thu Lng- n tp Cao Hc
-
Ng Thu Lng- n tp Cao Hc
)/1( 2x
-
Ng Thu Lng- n tp Cao Hc
-
Ng Thu Lng- n tp Cao Hc
-
+
1
cos dxx
x
Bi tp :Bi tp :
+ ln dxx 1)(xg =
Ng Thu Lng- n tp Cao Hc
2 2
dxx 2/3
)(x
xg =
-
20032003
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
-
20042004
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
-
20052005
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
1
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
20062006
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
20072007
-
3 21 1mdx
x x
+
+ 20082008
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
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
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
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 :
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
20201010
-
20112011)) Chng minh tch phn suy rng hi t v tnh tch phn
2 2( 1)( 3 1)x x x dx+ +
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
Ng Thu Lng- n tp Cao Hc
6 31 4 1
dxx x+
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
BBi ti tp p 11:: 21 1
dxx x
+
+ +
+
Tnh cc tch phn suy rng sau
Ng Thu Lng- n tp Cao Hc
BBi ti tp p 22:: 20
xx e dx+
+
2 2ln1 dx
xxBBi ti tp p 3 3 ::
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
+
++dx
xx 521
2BBi ti tp p 4 4 ::
Kho st s hi t ca cc tch phn sau :
Ng Thu Lng- n tp Cao Hc
BBi ti tp p 5 5 :: +
++12 1xx
dx
-
Chng II Chng II PhPhp p TTnh nh TTch Phnch PhnBBi i 1515 -- TTch Phn Suy Rch Phn Suy Rngng
BBi ti tp p 6 6 ::1 1
dxx
+
+
BBi ti tp p 7 7 :: 4 2( 1)x dx+ +
+ +
Ng Thu Lng- n tp Cao Hc
BBi ti tp p 7 7 :: 4 21 1x x
+ +