Đạo Hàm Và Vi Phân Cao Hoc

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


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


( 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

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