-
1
I. O HM V VI PHN
1. Cng thc o hm cc hm c bn
( ) ( )
1 1
2
2 2
( ).
( ) . ( ) ( )
1 ( )ln ln ( )
( )
sin cos sin ( ) ( ).cos ( )
cos sin cos ( ) ( ).sin ( )
1 '( )tan 1 tan tan ( )
cos cos ( )
arcsin
x x f x f x
n n n n
e e e f x e
x nx f x n f x f x
f xx f x
x f x
x x f x f x f x
x x f x f x f x
f xx x f x
x f x
x
2 2
2 2
1 ( )arcsin ( )
1 1 ( )
1 ( )arctan arctan ( )
1 1 ( )
f xf x
x f x
f xx f x
x f x
2. Cc quy tc ly o hm
2
( )
( . ) . .
. .
f g f g
f g g f f g
f g f f g
g g
( )f f
3. Cch tnh o hm ring ca hm 2 bin f(x,y) o hm ring theo x ( hoc y) : Coi y ( hoc x) l hng s
o hm cp 2 : ( ) , ( ) , ( ) ( )xx x x yy y y xy x y y xf f f f f f f
4. Vi phn
a. Hm 1 bin : 2 2
.df f dx
d f f dx
-
2
b. Hm 2 bin: 2 2 2 2
x y
xx yy xy
df f dx f dy
d f f dx f dy f dxdy
5. Cc dng bi tp a. Dng 1: Tnh ton o hm, vi phn ca hm 2 bin
Bi 1: Cho hm 2( , ) cos 2 sin(3 )f x y x y y x . Tnh (0, ) 3 (0, )
2 2xx yyA f f
22cos 18 sin(3 ), cosxx yyf y y x f x y 0, 0, 0, 0 02 2
xx yyf f A
Bi 2: Cho hm ( , ) ln( 2 )x yf x y e x y . Tnh (1, 1) 2 (1, 1)y xyA f f
2
2 2, ( )
2 ( 2 )
x y x yy xy y xf e f f e
x y x y
5 7 29
1, 1 , 1, 13 9 9
y xyf f A
Bi 3: Cho hm3 2( , ) 3 4f x y x y xy . Tnh (1,0)df ,d2f(1,0)
2
3 2 3 2
3 4 3 6 4 4(1,0) , (1,0) .
2 22 3 4 2 3 4
3(1,0) 2
2
x x y y
x y y xf f f f
x y xy x y xy
df dx dy
2 2 2 2 23(1,0) (1,0) (1,0) 2 7 24
xx yy xyd f f dx f dy f dxdy dx dy dxdy
Bi 4: Cho hm 3 3( , )f x y x y . Tnh df(1,1)
2 2
3 3 3 3
3 3 3 3(1,1) , (1,1)
2 2 2 22 2
3(1,1)
2 2
x x y y
x yf f f f
x y x y
df dx dy
Bi 5: Cho hm 2 2( , ) 1f x y xy x y .Tnh
2
(0,0), (0,0)f
dfx y
t 2 21u x y th .f xy u , ,x y
x yu u
u u
, (0,0) 0u
-
3
(0,0) 0, (0,0) 0x x y yx y
f yu xy f f xu xy fu u
(0,0) 0df
2 2
2 2
3
1. . (0,0) 1xy x y xyy
y
x xy x xf f u yu x y u x y f
u u u u u
Bi 6: Cho hm 2 2( , ) ln( )f x y y x y . Tnh
2
( 2,1), ( 2,1)f
dfx y
2 2
2 2
2 2
22,1 2 2
2ln( ) 2,1 2
2,1 2,1 2,1 2 2 2
x x
y y
x y
xf y f
x y
yf x y y f
x y
df f dx f dy dx dy
2 2 2 2 22 4
2,1 6 2( )
xy xy
x xyf y f
x y x y
Bi 7: Cho hm 2 2
( , ) 3 x yf x y e . Tnh 2
(1,1), (1,1)f
dfx y
2 2 2 2
2 2 2 2
2 2
2 2
6 6 (1,1) 6 ( )
12 6 .2 (1,1) 24
x y x yx y
x y x yxy xy
df f dx f dy xy e dx x ye dy df e dx dy
f xye xy x ye f e
Bi8: Cho hm 2 2( , ) ln( 3 )f x y x y . Tnh
2 2
2(1,1) (1,1), (1,1) (1,1)
f f f f
x y x y x
2 2 2 2
2 2 2
2 2 2 2 2 2
2 6, (1,1) (1,1) 2
3 3
2( 3 ) 4 12 1, ' (1,1) (1,1)
2( 3 ) ( 3 )
x y x y
xx xy xx xy
x yf f f f
x y x y
x y x xyf f f f
x y x y
Bi 9: Cho hm 2 2( , ) 6f x y x y . Tnh
2 2
2( 2,2) ( 2,2), ( 2,2) ( 2,2)
f f f f
x y x y y
-
4
2 2 2 2
2
3 2 2 32 2
6 3 2 1, 2,2 2,2
26 6
6 6 12 12 2 3 3 2' 2,2 2,2
64 64 16(6 )6
x y x y
yy xy yy xy
x yf f f f
x y x y
x xyf f f f
x yx y
Bi 10: Cho hm ( , ) sin1
xf x y
y
. Tnh
2
2 3 -5 f f f
x y x y
khi , 0
3x y
2 2 2
1 1 1cos , cos , cos sin
1 1 1 1 1 1(1 ) (1 ) (1 )
1 1 3 7 (5 3 3)2 ,0 3 ,0 5 ,0 1 3. 5 .
3 3 3 3 2 2 3 2 2 6
x y xy
x y xy
x x x x x xf f f
y y y y y yy y y
f f f
Bi 11: Cho hm 2 2( , ) ln(3 5 4 )f x y x y xy . Tnh 2 4 , df
f fA
x y
ti x=1, y=-1
2 2 2 2
6 4 5 10 4 7(1, 1) , (1, 1)
6 63 5 4 3 5 4
5 7 19 5 72 4 , (1, 1)
6 6 3 6
x x y y
x y y xf f f f
x y xy x y xy
dx dyA df
Bi 12: Cho hm 2
( , )2 5
x yf x y
x y
. Tnh
2 2
2 22 3 , B=4 -2
f f f fA
x y x y
khi
x=-2, y=-1
2 2
3 3
9 9( 2, 1) 9, ( 2, 1) 18 36
(2 5 ) (2 5 )
4.9 10.9( 2, 1) 36, ( 2, 1) 180 216
(2 5 ) (2 5 )
x x y y
xx xx yy yy
y xf f f f A
x y x y
y xf f f f B
x y x y
Bi 13: Cho hm 2 24 4 1( , ) x xy yf x y e . Tnh 2 3 ,
f fA
x y
ti (x,y)=(3,2) v
2 2
2 2B=3 -2 ,
f f
x y
ti (x,y)=(1,1)
-
5
2 2 2 2
2 2 2 2
4 4 1 4 4 1
2 4 4 1 2 4 4 1
2 2
2 2
(2 4 ) , (8 4 )
2 (3,2) 3 (3,2) 2( 2) 3.4 8
f = 2+(2x-4y) . , = 8 (8 4 ) ,
B=3 (1,1)-2 (1,1) 3.6 2.24 30
x xy y x xy yx y
x xy y x xy yxx yy
f x y e f y x e
f fA
x y
e f y x e
f f
x y
-
6
b. Dng : Cc tr ca hm 2 bin f(x,y)
Cch lm
Bc 1: Tm im dng : Bng cch gii hpt 0
0
x
y
f
f
. Gi s ta c cc im dng
( , )i i iM x y
Bc 2: Tnh 3 o hm ring cp 2.
Xt ti tng im dng Mi(xi,yi) bng cch t
2( , ), ( , ), ( , ), xx i i xy i i yy i iA f x y B f x y C f x y B AC
0: Hm khng t cc tr ti Mi ang xt
0: ( , )0
0 : ( , )
ct i i
cd i i
A f f x y
A f f x y
0 : Khng xt
Bi tp: Tm cc tr cc hm sau
3 2 2 2
2
2 3
3 3
3 2
1/ ( , ) 2 5
2 / ( , ) 3 8ln 6ln
3 / ( , ) 3 18 30
4 / ( , ) 6
5 / ( , ) 12 8
f x y x xy x y
f x y x xy x y
f x y x y y x y
f x y x y xy
f x y x y x y
3 3 2
2 2
3
3 3
4 2 2 3
6 / ( , ) 3 3 1
7 / ( , ) ( 2 )
8 / ( , ) 12 48
9 / ( , ) 8 6
10 / ( , ) 2
x
f x y x y y x
f x y x y y e
f x y x xy x
f x y x y xy
f x y x x y y y
-
7
c. Dng 3: Gii phng trnh vi phn cp 2 : 1 2 ( )y a y a y f x
Ta tm nghim tng qut ca pt di dng tq tn ry y y
Trong ytn l nghim ca pt thun nht
V yr l 1 nghim ring ca pt ban u
PT c trng 2
1 2 0k a k a
o Pt c 2 nghim thc phn bit : 1 21 2 k x k x
tny C e C e
o Pt c nghim kp : 1 2 kx kx
tny C e C xe
o Pt c nghim phc : 1 2 k= : ini cos sx x
tny C e x C e x
Nghim ring ca pt ban u (ch xt trng hp c bit khi f(x) c dng
( ) ( ( ).cos ( )sin )x n mf x e P x x Q x x
Khi , nghim ring yr c dng
( ( )cos ( )sin )h s sx
ry x e T x x R x x
Trong s=max{m,n},
0 nu h i khng l nghim pt c trng
1 nu h i l nghim n ca pt c trng
2 nu h i l nghim kp ca pt c trng
Sau , tnh h cp 1, cp 2 ca yr v thay vo pt ban u tm c th 2 a thc Ts(x),
Rs(x)
Bi tp: Gii cc ptvp sau
2
2
2
3
2
3
3
1. 3 2
2. 5 6 3
3. 5 6 5cos2
4. 7 6 3 2 1
5. 5 4 5sin 2cos
6. 4 3 3
7. 4 4
8. 2 3 (2 1)
9. 6 9 2
10. 3 2 3cos
x
x
x
x
x
x
y y y xe
y y y e
y y y x
y y y x x
y y y x x
y y y e
y y y e
y y y x e
y y y xe
y y y
2sinx x
-
8
Gii: 2
2
21 2
1. 3 2
: 3 2 0 1, 2
x
x xtn
y y y xe
ptdt k k k k
y C e C e
V phi : ( ) .cos0 0sin0x xf x xe e x x x
Suy ra ( )xry xe ax b . Ta tnh o hm v thay vo pt cho tm a, b
2 2
2 2 2
( 2 ), ( 2 2 2 )
( 4 2 2 ) 3 ( 2 ) 2 ( )
1( 2 ) (2 ) , 1
2
x xr r
x x x x
y e ax bx ax b y e ax bx ax b ax a b
e ax ax bx a b e ax ax bx b e ax bx xe
a x a b x a b
Vy
nghim pt l 2
1 2
1( )2
x x x xtn ry y y C e C e e x x
2
2
2 31 2
2 2
2 2
2 2
2 2 2 2
2. 5 6 3
: 5 6 0 2, 3
( ) 3 (( 3).cos0 0.sin 0 )
( cos0 sin 0 )
(2 1), (4 4)
(4 4) 5 (2 1) 6 3
35 3
5
x
x xtn
x x
x xr
x xr r
x x x x
tn
y y y e
ptdt k k k k
y C e C e
f x e e x x
y xe a x b x axe
y ae x y ae x
ae x ae x axe e
a a
y y
2 3 21 23
5
x x xry C e C e xe
-
9
2
2 31 2
0
3. 5 6 5cos2
: 5 6 0 2,3
( ) 5cos2 5.cos2 0.sin 2
cos2 sin 2
2 sin 2 2 cos2 , 4 cos2 4 sin 2
4 cos2 4 sin 2 5 2 sin 2 2 cos2 6 cos2 sin 2 5cos2
x xtn
x
r
r r
y y y x
ptdt k k k
y C e C e
f x x e x x
y a x b x
y a x b x y a x b x
a x b x a x b x a x b x x
2 31 2
2 10 5 5 25,
10 2 0 52 52
5(cos2 5sin 2 )
52
x x
a ba b
a b
y C e C e x x
2
2
61 2
2 0 2
2
2 2
4. 7 6 3 2 1
: 7 6 0 1,6
( ) 3 2 1 (3 2 1)cos0 0.sin 0
, 2 , 2
2 7(2 ) 6( ) 3 2 1
6 31 3
14 6 2 , ,2 2
2 7 6 1
x xtn
x
r r r
y y y x x
ptdt k k k
y C e C e
f x x x e x x x x
y ax bx c y ax b y a
a ax b ax bx c x x
a
a b a b c
a b c
6 21 2
17
12
16 18 17
12
x xy C e C e x x
-
10
II. KHO ST HM y=f(x). CHUI S - CHUI LU THA
1. Mt s gii hn thng gp
1
( )
11 1 0
11 1 0
0
( ) 0
0,
...lim ,
...
,
lim , lim 0
lim ln , lim ln
1lim 1 lim 1 ( )
f n
n nn n n
m mx mm m
x x
x x
x x
n
n f n
m n
a x a x a x a am n
bb x b x a x b
m n
e e
x x
e f n en
2. Dng 4: Kho st hm s y=f(x) 1. Tm MX:
Lu : + Hm f(x) c cha mu s th mu s c iu kin khc 0 + Phn trong cn bc chn phi c iu kin ln hn hay bng 0 + f(x) c cha hm ln(g(x)) th phi c ii kin g(x)>0
2. Tm cc tr - GTLN, GTNN bng cch gii pt f(x)=0
Nu f(x)>0 trong khong (a.b): hm tng trong khong (a,b)
Nu f(x)0 th hm t cc i v f(x0)
-
11
TCX
lim
l
( )
( )
( )
im
lim
x
x
x
f x
fkh
xy ax b a
x
f x ax b
i
4. Tm im un, khong li, lm : Tnh h cp 2
Nu f(x)>0 trong khong (a,b): hm lm trong (a,b)
Nu f(x)
-
12
3. Tm khong li , lm v im un ca cc hm
3 2
2
3 6 2
2
1
4
y x x x
xy
x
y x x
4. Tm GTLN, GTNN trong cc on tng ng
2
2
2 ,( , )
32ln , [ , ]
2
( 3) , [ 1,4]x
y x x
y x x e
y x e
-
13
3. Dng 5: Kho st s hi t ca chui s
Cho chui s
1
, 0n nn
u u
a. iu kin cn ca s hi t: Nu lim , 0nn
u a a
hoc lim nn
u
th chui
1
n
n
u
Phn k
b. Tiu chun Cauchy : Gi s lim | |n nn
u C
th 1:
1:
C HT
C PK
Ch : Khng xt trng hp C=1. Tuy nhin, khi un>0, C=1 v 1,n nu n th ta vn kt lun
chui PK
c. Tiu chun dAlembert: Gi s 1| |
lim| |
n
n n
uD
u
th
1:
1:
D HT
D PK
Ch : Khng xt trng hp D=1. Tuy nhin, khi un>0, D=1 v 1 1,n
n
un
u
th ta vn kt lun
chui PK
Bi tp : Kho st s hi t ca cc chui sau
2
2 11
1
1
3 2
1
( 1)
1
1.3.5...(2 1).31 /
2 . !
!32 /
2.4...(2 )
1 13 / (1 )
3
3 14 / ( )
7 2
1 15 / (1 )
3.4
n
nn
n
n
n
nn
n
n
n n
nn
n
n
n
n
n
nn
n
n
2
1
2
1
1
1
22
0
(2 1)!!6.
3 !
5 ( !)7.
(2 )!
1 18. 1
2.5
19.
14
210.
3 1
nn
n
n
n
nn
n
nn
n
n
n
n
n
n
n
n
n
nn
n
-
14
4. Dng 6: Tm min hi t ca chui lu tha
Dng 0lim , lim ( )n n
n nn n
a x a x x
. Ta c th t x-x0=X a chui sau v ng dng chnh tc
nh chui trc. Do vy, ta thng nht rng dng chnh tc ca chui lu tha l lim nnn
a x
tm
Min hi t ca chui lu tha, ta lm qua 2 bc sau
Bc 1: Tm Bn knh hi t (BKHT) R . t 1
lim | |
| |lim
| |
nn
n
n
n n
a
a
a
th
0,
, 0
1 ,0
R
Bc 2: Xt ti x=R v x=-R, tc l k.st s HT ca 2 chui s lim , lim ( )n nn nn n
a R a R
Bc 3: Kt lun v MHT tu theo kt qu bc 2. Nu c 2 chui trn HT th MHT l [-R,R]
Nu 1 chui HT, 1 chui PK th MHT l [-R,R) hoc (-R,R]
Nu 2 chui PK th MHT la (-R,R)
Ch : Khi kho st s HT ca 2 chui s ng vi X=R, X=-R , ta lm theo th t u tin sau
1. Nu chui s sau khi rt gn c dng lim , 0n
cc
n th
, 1
, 1
HT khi
PK khi
2. Nu chui s sau khi rt gn c dng ( 1)
lim , 0n
n
cc
n
th
, 0
, 0
HT khi
PK khi
3. Nu khng c 2 dng trn th tnh gii hn lim nn
u
v ta c kt qu sau
lim 0nn
u c
lim nn
u
th chui PK.
Lc c 2 chui s ng vi x=R v x=-R u PK
4. Nu khng c 2 dng u v lim 0nn
u
th dng t/c Cauchy hoc dAlembert v lm theo 2 ch ca
2 t/c trn.
Bi tp
Tm MHT ca cc chui lu tha sau
-
15
2 61
11
2
1
1
1
1
1
3 21
( 2)( 1)7 /
5 . 1
( 1) ( 3)8 /
2 (2 1)
19 /
4 1
( 1)10 /
(2 1)(3 4)
211/
2 1
( 1)12 /
1( 1)
n
nn
n n
nn
nn
n
n n
n
n n
n
n n
n
n x
n
n x
n
nx
n
x
n n
x
n
x
n n
2
1
1
1 1
1
31 4 21
2
1
1
2 31
41/ ( 2)
2 1
( 1)2 /
.2 . 1
( 1) 2 ( 5)3 /
( 1) ln( 1)
( 1) ( 2)4 /
3 1
( 1)5 /
4 (3 1)
( 1) 36 / ( 1)
4 1
nn
n
n
nn
n n n
n
n n
nn
n n
nn
n nn
nn
nx
n
x
n n
x
n n
x
n n
x
n
xn
-
16
n cho t n tp th nht
1
Cu 1: Gii pt 25 4 ( ) xy y y x x e
Cu 2: Cho hm 2( , ) arctan 2y
f x y x yx
. Tnh 3x yA f f v xx xyB f f ti x=1, y=0
Cu 3: Tm cc tr hm 8
( , )x
f x y yx y
vi x>0, y>0
Cu 4: Tm tim cn v cc tr hm
2
1 xy xe
Cu 5: Kho st s hi t ca chui s 5
2
1
11
2 1
n
n
nn
n
Cu 6: Tm min hi t ca chui lu tha 1
1
3( 1)
3 1
nn
n
xn
2
Cu 1: Gpt 5 6 2siny y y x
Cu 2: Kho st s hi t ca chui s 2
1
2 ( !)
(2 )!
n
n
n
n
Cu 3: Tm min hi t ca chui lu tha ( 1)
1
( 1) 1
2 15
nn
nn
x n
n
Cu 4: Cho hm ( , ) 3 cosx yf x y e x y . Tnh
(0,0) 4 (0,0), ( , ) 2 ( , )4 4 4 4
x y xx xyA f f B f f
Cu 5: Tm cc tr v tim cn ca hm 2 2 1
2
x xy
x
Cu 6: Tm cc tr hm 2 2( , ) 2 3f x y x y xy y x
3
Cu 1: Gpt 23 2 2 5y y y x x
-
17
Cu 2: Kho st s hi t ca chui s 1
( 1) 2
3 5
n n
n nn
Cu 3: Tm min hi t ca chui lu tha 2
3 21
( 1) ( 2)
2 2
n n
n
x
n n
Cu 4: Cho hm 2( , ) ln( 2 )f x y x xy y . Tnh
3 (1,0) 5 (1,0), ( 1,0) 2 ( 1,0)x y xy yyA f f B f f
Cu 5: Tm cc tr v tim cn ca hm 2 2 1
2
x xy
x
Cu 6: Tm cc tr hm 3 3( , ) 6f x y x y xy
4
Cu 1: Gpt 2 (2 1) xy y y x e
Cu 2: Kho st s hi t ca chui s ( 1)2
1
3 ( 1) 2 1
2 15
n nn
nn
n n
n
Cu 3: Tm min hi t ca chui lu tha 3 2
1
2 ( 1)
2 2
n n
n
x
n n
Cu 4: Cho hm 2( , ) 2 1yf x y xe y . Tnh
2 (2,0) (2,0), (5,0) 2 (5,0)x y xy yyA f f B f f
Cu 5: Tm cc tr v tim cn ca hm 3
2
1
1
xy
x
Cu 6: Tm cc tr hm 3 3( , ) 27 27f x y x y xy
-
18
TCH PHN
I. Nhc li tch phn
1. Tch phn 1 s hm c bn 1
2
2
1
, 0ln
ln
sin cos
cos sin
1tan
cos
1cot
sin
xx
xx dx C
aa dx C a
a
dxx C
x
xdx x C
xdx x C
dx x Cx
dx x Cx
2 2
2 2
2 2
2 2
2 2
2 2
1 1arctan
1arcsin
1ln
1ln
dx x Cax a
xdx C
aa x
dx x a x Ca x
dx x x a Cx a
2. Cng thc Newton Leibnitz
Gi s ( ) ( )f x dx F x C (ta gi F(x) l 1 nguyn hm ca f(x)) th
( ) ( ) ( )b
a
f x dx F b F a
3. Hai phng php tnh tp
a. Phng php i bin
t ( ) ( )x t dx t dt th ( ) ( ( )) ( )f x dx f t t dt
Hoc t ( ) ( )t x dt x dx nu c th bin i ( ) ( ( ))f x g x
b. Phng php tp tng phn
( ) ( ) ( ) ( ) ( ) ( )u x v x dx u x v x v x u x dx
c. Bi tp
21
2
2
32
ln8
40
2 1
2 3
2
2
9
1x
I x x dx
xI dx
x
xI dx
x
dxI
e
25
26
7
8
29
2310
(2 1)
( )
sin 2
(3 2)cos2
( 2 ) ln
(3 4 )
x
x
x
I x e dx
I x x e dx
I x xdx
I x xdx
I x x xdx
I x x e dx
11
22 2
121
3
13 21
4
14 22
( 1)arctan
ln( )
1
5 6
6 5
e
I x xdx
I x x dx
xI dx
x x
xI dx
x x
-
19
II. Phng trnh vi phn cp 1
1. Pt tch bin
a. Dng ( ) ( )f x dx g y dy
b. Cch gii: Ly tp 2 v pt cho
( ) ( )f x dx C g y dy
2. Ptvp tuyn tnh cp 1
a. Dng ( ) ( )y p x y q x
b. Cch gii: S dng trc tip cng thc tnh y
( ) ( )( )p x dx p x dxy e q x e dx C 3. Ptvp ton phn
a. Dng ( , ) ( , ) 0P x y dx Q x y dy vi iu kin y xP Q
b. Cch gii: Chn 0 0( , )x y sao cho cc hm P, Q xc nh ti , th dung CT
0 0
0( , ) ( , )yx
x y
P x y dx Q x y dy C
4. Bi tp
2 2
2 2
2 2
2
3
2 3 2
2
1. 1 1 0
2. tan sin cos cot 0
3.( 1) ( 2) 0, (0) 1
4. tan sin
5.(1 ) 4 3
3 26. , (1) 0
7. 3
8.(3 4 ) 3 0
29. 5
10.( sin ) (cos ) 0
x
y y
y dx y x dy
x ydx x ydy
x dy y dx y
y y x x
x y xy
y y yx x
yy xe
x
x y x dx xy dy
yy x
x
e x dx y xe dy
-
20
32
4
3
2 2
2 3
3
3 2 4
3
11. 0, (4) 22
412. cos
3 6sin13. , 0
14.(5 4 ) (5 4 ) 0
315. 2 , 0
16.( sin 5 ) ( cos 5 ) 0
17. 3 4 , (0) 2
18. 4 (8 12 10 3)
19. 3 4
x
x x
x
x
x
ydx x dy y
yy x x
x
y xy x
x x
xy y dx x y x dy
yy e x x
x
e y y dx e y x dy
y y e y
y y x x x e
y y e
3
cos5
320. cos(4 )
x
yy x x
x
III. Tch phn kp
Ch xt 1 loi tch phn trn min D l 1 phn hnh vnh khn
( , )
( , )D x y
I f x y dxdy
2 2 2 2 2 22 1 1 1 2 2( , ) : , , ,D x y x y r x y r a x b y a x b y i bin bng cch t cos , sinx r y r th
( , )
. ( cos , sin )D r
I f r r drdr
Khi , ta xc nh cn tch phn nh sau
1. i vi r : 1 2 1 2r rr r r
2. i vi : ta cn c vo 2 ng thng a1x=b1y, a2x=b2y tnh 2 gc
1 2 1 2
Cc trng hp c bit
a. Nu khng c 2 ng thng: 0 2
b. Nu ly na pha trn hnh vnh khn: 0
c. Nu ly ng vi x>0, y>0 : 02
-
21
Cui cng, thay cn tch phn tm vo tp ban u
2 2
1 1
. ( cos , sin )r
r
I d r f r r dr
Bi tp: Tnh cc tp sau vi min D tng ng
2 2
2 2
2 2 2 21
2 22
2 2
2 2 2 23
2 24
2 2 22 2
52 2
2 26
( ) , :1 9, 0
1, : 4, 0, 0
, : 4, 0
, : 1, 0
cos, :
4
, : 1, 0
D
D
D
x y
D
D
x y
D
I x y dxdy D x y x
I dxdy D x y x yx y
I x y dxdy D x y y
I e dxdy D x y x
x yI dxdy D x y
x y
I e dxdy D x y x
2
2 27
2 2 2 28
2 2 2 29
2 2
2 2 2 210
2 2 2 2 2 2 211
12
1, : 4,0
4
4 , : 4,0 , ,
1, : 1, 33,0 ,0 , , 3
3
arctan , : 3
ln( ) , :1
D
D
D
D
D
x
I dxdy D x y x y
I x y dxdy D x y y y x y x
I dxdy D x y x y x y y x y xx y
I x y dxdy D x y
I x y x y dxdy D x y e
I e
2
2 2 2
2 22 2 2 2
13
, :1 ln 3
25sin( ) , :
36 36
y
D
D
dxdy D x y
I x y dxdy D x y
Bi kim tra
-
22
1. Gpt :
2
1. tan sin , ( ) 04
2.(2 sin ) (2 cos ) 0
3. 6 5
y y
x
y y x x y
e y dx xe x y dy
y y y xe
2. KS s HT ca chui 2
1
( 1) (2 1)!
3 ( !)
n
nn
n
n
3. Tm MHT 3 5 31
(2 3)
2 1
n
n
x
n n
4. Tm cc tr v tim cn 3 3 22y x x x
5. Cho ( , ) 2 3 lnx yf x y e y x . Tnh (1, 1), (1,0) (1,0)xx yyA df B f f
IV. Tch phn suy rng loi 1
1. Tch phn c bn
1,
1
adx
x
HT khi
PK khi
2. Lu : Ch xt hm f(x)>0 trn on ly tch phn
Bi tp:
KS s HT ca cc tp sau
3 22
4 3 21
1.ln
2.2 5
3.1
e
dx
x x
dx
x x
dx
x x x x
-
23
1
31
22
21
21
232
2
31
4.2 3sin
5.3ln
( 2)6.
1
sin7.
2ln
8.cos2
9.(2 )(1 )
( 1)10.
5 2
dx
x x
xdx
x x
x dx
x x
xdx
x x
dx
x x
dx
x x
x dx
x x
Tm m tich phn HT
2 21
21
2
5 31
0
3 2 21
1.( sin )
2.2
( 1)3.
4 2
arctan4.
(1 )
5.( ln(1 ))
m
m
m
m
m
dx
x x x
dx
x x
xdx
x x
xdx
x
dx
x x x
-
24
V. Tch phn ng loi 2 khng ph thuc ng i
1. Dng ( , ) ( , )C
I P x y dx Q x y dy , C l ng cong cho trc
2. iu kin : x yQ P
3. Cch tnh : Tm hm U(x,y) sao cho dU=Pdx+Qdy tc l
C C
I Pdx Qdy dU , trong hm U c tm bng cch chn (x0,y0) sao cho 2
hm P, Q v cc o hm ring ca chng xc nh ti (x0,y0)
0 0
0( , ) ( , )yx
x y
U P x dx Q yy x dy (Lu : Tp theo dx th y l hng s)
Khi , tp ch cn ph thuc vo im u A v im cui B ca ng cong C v ta c kt
qu
( ) ( )C AB
I Pdx Qdy dU U B U A
Bi tp
1. Cho 2 hm , ( , ) 2 cos , ( , ) 2 sinxy mx xy mxP x y ye e y Q x y xe e y
a. Tm m Pdx+Qdy l vi phn ton phn ca 1 hm U(x,y) no . (S: m=1)
b. Tnh tp
C
I Pdx Qdy bit C l ng cong tu i t A(0,) n B(1,0) (S: I=1+e)
2. Cho 2 hm 2 3( , ) 3 7, ( , ) 2m nP x y x y Q x y x y
a. Tm m, n sao cho Pdx+Qdy l vi phn ton phn ca 1 hm U(x,y) no . (S: m=2, n=1)
b. Vi m, n trn, tnh tp ( , ) ( , )C
I P x y dx Q x y dy vi C l ng parabol y=x2+x+1 i t
A(1,3) n B(-1,1) (S: I= - 24)
3. Cho 2 hm P(x,y)=y,Q(x,y)=2x-yey
a. Tm hm h(y) tho h(1)=1 sao cho tch phn ( ) ( , ) ( ) ( , )C
I h y P x y dx h y Q x y dy khng
ph thuc ng i
b. Vi hm h(y) trn, tnh tp I bit C l ellipse 4x2+9y2=36i t A(3,0) n B(0,2)
4. Cho 2 hm ( , ) (1 ) , ( , ) (1 )y yP x y x y e Q x y x y e
a. Tm hm h(x) , h(0)=1 sao cho h(x)P(x,y)dx+h(x)Q(x,y)dy l vi phn ton phn ca hm U(x,y)
no
-
25
b. Vi hm h(x) tm c, tnh tch phn ( ) ( , ) ( ) ( , )C
I h x P x y dx h x Q x y dy vi C l na
ng trn x2+y2=9, x>0 i t A(0,-3) n B(0,3)
5. Cho 2 hm 2 3 2( , ) , ( , ) (1 )P x y x y Q x y x y
a. Tm hm ( , ) m nh x y x y , m, n l hng s sao cho h(x,y)P(x,y)dx+h(x,y)Q(x,y)dy l vi phn
ton phn ca hm U(x,y) no .
b. Vi h(x,y) va tm, tnh tp ( , ) ( , ) ( , ) ( , )C
I h x y P x y dx h x y Q x y dy vi C l ng cong
tu i t
A(1, )2
n B(0,1)
6. Cho 2 hm 2 2( , ) 3 , ( , )P x y xy y Q x y x xy
a. Tm hm 1
( , )( )
h x yxy mx ny
m, n l hng s v h(1,1)=1/3 sao cho
h(x,y)P(x,y)dx+h(x,y)Q(x,y)dy l vi phn ton phn ca hm U(x,y) no .
b. Vi h(x,y) va tm, tnh tp ( , ) ( , ) ( , ) ( , )C
I h x y P x y dx h x y Q x y dy vi C l ng cong
x2+2xy-y2+5x-5y=2 i t A(1,1) n B(2,3)
HNG DN GII BT tp ng loi 2
3.
a. Tm h(y) tc l hm h(y) ch theo 1 bin y, khi 0,x yh h h .
tp ( ) ( , ) ( ) ( , )C
I h y P x y dx h y Q x y dy l tp khng ph thuc ng i ta phi c
( ) ( , ) ( ) (, )y x
h y P x y h y Q y
. .1 0.(2 ) .2
.
( )
1
1
ln | | ln | |
yh y h x ye h
h y h
dhy h pt tach bien
dy
dhdy
h y
dhdy C
h y
h y C
Thay iu kin h(1)=1 vo ng thc trn, ta c C=0. Vy h=y
Khi , thay h(y)=y vo tp cn tnh, ta c 2 (2 )y
AB
I y dx x ye ydy
-
26
C 2 hm y2 v (2x-yey) u xc nh vi mi x, y nn ta chn (x0,y0)=(0,0) v tnh hm U
2 2
0 0
( , ) (2.0 )yx
y y yU x y y dx ye dx y x ye e
Vy 2(0,2) (3,0) 1I U U e
2 3
3 3 2 2 4
2
2 2 4 3 3 2
2
3 2 2 4 4 3 3 2
3 2 3 3 2 2 4 1
3 2 3
3 2 ln 2ln
1 16 2 6 2
2 13 6 3 6 2
16 2 , 3 6 2 ,
12 8 6 2
12 18 4
x
y
m n
m n
m n m n
m n
U x y xy x y
u xy y x y x y xyx x y
u x xy x y x y xy x y
P x y x y xy Q x y x y x h x yx y
x y x y x x y n x y x y xy x y
x y x y x x y
4 3 3 2 1
2 3 1 1 3 2 2 4 4 3 3 2
3 4 2 4 2 3 4 2
3 6 2
10 5 6 2 3 6 2
10 5 (6 3 ) (2 6 ) ( 2 )
2 , 3 5, 2 5
m n
m n m n
m x y x y x x y
x y x x y x y nx x y x y xy my x y x y x
x y x y x y n m x y n m x y n m
n m n m n m
