Phương pháp tính tích phân và số phức phần 1

HO Chi JViinhJ PHUONG PHAP TINH... Vay fx khon.

515.076

iGV cbuyen Toan Trung tam luyen thi Vinh Viin - TP HO Chi JViinhJ

PHUONG PHAP TINH

(Tdi ban idn thii nhat, c6 siia chita vd bo sung)

N H A X U A T B A N D A I H O C Q U O C G I A H A N O I

T ] T i n h dao ham cua F(x) = x.lnx - x, roi suy ra nguyen ham ciia f(x) = Inx

Gidi

T a CO : F(x) = x.lnx - x = x(lnx - 1)

Suy r

a : F'(x) = [x(lnx -

Vay the

o din

h nghI

a cu

a nguye

n ha

m, nguye

n ha

m cii

a f(x) =

x +

C

~2\h dao ha

m cii

a F(x ) = x^lnx, ro

n F'(x) =

2xlnx

+ — x^ =

2xlnx

+ x = f(x

) +

x

Vay ' F

'(x)dx=

— +

C = x^lnx - — +

Vay mo

t nguye

n ha

m cii

a f(x) l

a F(x) = xllnx

~3\h nguyen

h am ciia f(x) - xVx +

1 bie

t F(0 ) =

x +

1

Vay

= ( X

= f

{x + l

)2 dx -

("(x + l

x + 1 )- f (X

+ l

)2 d(

x

+ l)

2

2

-= -(

X

+

l)

2 (X + 1) 2

+

C

5 3

= -(

-x +

l +

C

Hay F(x) =

(x + l)V

nen ta

CO :

2

= (0 + 1)^

Vo +

1

- (

0 + 1)V0 + 1

(x + D^V

-x + l (x + l)V

2 Chufng m i n h r k n g G(x) = - ( 1 + x)e"'' la mot nguyen h a m cua

g(x) = x.e'" Roi suy ra nguyen h a m cua k(x) = (x - De""

Gidi

1 Ta CO : F'(x) = e" + e^Cx -2) = e^lx - 1) = f(x)

Vay F(x) la mot nguyen h a m cua f(x)

2 Ta CO : G(x) = - ( 1 + x)e"''

Suy ra : G'(x) = -e"" + (1 + x)e-'' = e"\ = g(x)

Vay G(x) la mot nguyen h a m ciia g(x)

Suy ra nguyen h a m cua k(x) = (x - l)e~'' = xe"" - e"'' = g(x) - e"

N e n k(x)dx = Jg(x)dx - je-^dx = G(x) + e"" + C

Vay nguyen h a m cua k(x) \k K(x) = -xe"" + C

~G\h da

o hk

m cu

a (p(x ) = (ax + b)e'' Ro

Gidi

TiX

gia thie

t (p(x ) =

(ax + b)e''

Suy r

a : (p'(x) =

(a +

ax + b)e''

a cho

n a = -

1, b = 1

Thi <p(x ) =

(-X + De" =

> (p'(x ) =

a F(x ) = (-x + De" +

C

~T\g min

h F(x ) = In| x + Vx

^ +

K | la mot nguye

n ha

m cu

a

fix) = , ^

tren R

Vx

^+

K Gidi

x + Vx^

+ K

Vx^

+ K(

n R

sl

Cho ha

m s o

o ha

m so ' F(x ) = (ax^ +

bx + c)

Dai hoc

Su pham

Ki thudt TP.HCM

Gidi

Ta

C O : F'(x) =

(2ax + b)

V3

-x

(ax^ +

bx + c)

2V3-X

1

2

2V3-X

1 [(2ax + b).2(3-x) -

(ax' +

bx + c)]

[-5ax

^ + (12a - 3b)x + 6b -

i x < 3

o F'(x ) =

c = 2(3 - x)x V

x <

3

6

Vay F'(x) la mot nguyen h a m cua f(x) vdi Vx ^ ± a

1 neu x = 0 ( x - D e ^ + l

F'(0) = lim ^ ^

= lim ^ 2x o x^ .0 X-

(Quy t^

c L'Hopital)

TiT (1), (2), t

a su

y r

a F(x) l

a nguye

n ham cua f(x) tre

n R

Vay FXO) =

lim — =

- = fTO) 2 2 o x^

(2)

IT] Tin

h da

o ham cua F(x) = (x^

- l)ln

Suy ra nguyen

ham cua f(x) = x

: F(x) = (x^

- l)ln

11

+ x | - x^ln | x

I ^-1 x 1 + + 1 I x 2xln F'(x) = ra Suy

x +

1

- 2xln IX

-= 2xln IX + 1

i

1 2xln | x | v6

-i x

0, x ^ 1

Taco: f(x) = xln

l

fl + x^ il

1X1

]

= 2xln

11

+ X

Suy ra ff(x)dx =

[F '(x)d

x + fldx =

ham cua f(x) = x

- Dln

ll + x

| -x

^lnl

o F(x) = e""^

(atan^x + btanx

+ c) l

tan^x tr

x + btanx

+ c) + e''^

[2a(

l + tan^

x)tan

<=> e 2atan^ x + (yl2a + b ) t a n 2 x + (^^b + 2a)tanx + (V2c + b)

(do quy t i c L'Hopital)

(1)

(2)

im

x -»o-

x

-0

X

FX O- )=

-l -l im l^

^^

x->0

" X -1

^ F'iOl

=

1

l im

1

Vay

F'(0*

) = F'(0"

y r

a F(x ) l

a nguye

n ha

m cii

a f(x ) tre

n R

(3)

14

I Churngmin

h F(x ) =

— I

n x 2 4

(X >

X

(x >

0)

0

(x =

0)

Dai hoc Yduac TP.HCM

x + = xln

ta

CO

: F(O^

) = l im ^^

" "^

^^

^ ^ = li

* X

= lim — I n

X

- li

m — = l im -

0

x-^O

* 2 x-

yO

* 4 x^

.0

* 2

(1)

= li

= li

m x^o^

V 2)

Tix

(1), (2 ) t

a ke

t lua

n F(x ) l

a nguye

r

= 0

= f(x )

(2)

I hkm

cua fix ) tre

^x^ + 2

^R

+ lj

x^-1 xUl

10

- 2 x + C v d i X < - 1 x^ + C v d i - 1 < X < 1 2x + C v d i X > 1

I Ti

m h

o nguye

n ha

m cii

a f(x) = | x |

-xdx neu

x <

0 + C

neu

X >

0

+ C neu

X <

19

I Ti

m h

o nguye

n ha

m cii

a f(x) =

x i x |

Ta

CO

: Jx

-x^dx neu

m cii

a f(x) = (x +

\x\f

Ta

CO :

Gidi

(x+

I X

I )2dx = Jlx

^ + 2x

I X I +x2 )dx

' 2

4x 4x d

x = + C

X <

0

21 I Tim ho nguyen

ham ciia f(x) = cos

X

Dai hoc Yduac TP.HCM

-2001 -He nhdn

Gidi

Ta

CO

: F(x) =

f

• CO

S xd

x ' d(si

n x)

22I Ti

m h

o nguye

n ha

f(x)dx =

Gidi

2

\3 2\

53

MX = f

(2

.32.5^)''d

x = 2250"

hi(2250)

+ C

1 2

23 I Tim ho nguyen ham cua :

25 \m ho nguyen ham cua f(x) = ——

28] Ti

m h

o nguye

n ha

m cu

a fix) = Ve" +

e

2

DH Y Thai Binh

2 =

e2

- e

2 e2

- e

2

f(x) = f(x) =

X — > neu — 2 2

' X

X — < neu — 2 2

ho nguye

n ha

m cii

a f(x) -Inex

1

+ xln

(1

+ xlnx)'d

x = (l.ln

x + — x)d

x X

= (In

x + l)dx = Inex.d

x

Vay f(x)dx =

Inex.dx rd(l + xlnx)

x

f (x)d

x = I

ho nguye

n ha

m cu

a f(x) = x(l - x)-°

DH Quoc gia Ha

1) +1](

1

- x)^

" = (x -

Nen f ( x ) d x = ( x - l ) 2 M x + ( x - l ) 2 ° d x

( x - l ) ^ M ( x - l ) + ( x - l ) 2 ° d ( x - l ) = ( X - 1 ) 2 2 ( x - l ) 2 1

, 2 0 0 1

DH Quoc gia Ha Ngi - 2000

b) Dat u = x^ + 1 So sanh hai ket qua t i m ducfc

Dai hoc Tong hap TP.HCM ~ A/1977 Gidi

a) Dat X = tana => dx = (tan^a + l)da, thi :

= — sin* a + C = - (tan* a cos* a) + C

b)

Da

t

u = + 1

=>

du =

-2 -3 o.

1

1 l-2u,^

l(

x^

+2x^

+1

4(x2

+

if

+ C + -

+ C

-

Ci

- C,

Ii

I2

+ C

- Ci

33

I T

im ho nguyen

ham cua

fix) =

Vx^ + x"" +

2

Gidi

Ta

CO :

f(x)dx =

I x2

+ X-2

I

Vx^ +x-^

+2

<ix = V(x2+X

-*

1

\ + x -

^ ) d x

=hilx

l — + C

-= Inix

t , = ln(x.Vx2

+3) +

C Tin

h F(x) = fVx2 + 3dx

DHYHaNoi -1999

Gidi

Fix) =

[Vx2 + 3dx = xVx2 +

3 jx

-d(Vx2 + 3) (Tic

h phan

- fV77

-(x) +

31n(xVx2 +3) +

(xV(x

2 + 3) +

35 I T i m ho nguyen h a m cua fix) =

-l

(x^ + 5x + IXx^

- 3

x +

1)'

DH Quoc gia Hd

Ngi A/2001

-Giai

Ta

CO :

x^

-l

(x^ + 5x + IXx^

- 3

x +

1) 1

2x +

5 1

2x-

1

F(x)=

8 2x + 5

J

- 1 5x + X +2x -

3 ,

-In 1 3x + X x-" -

ho nguye

n ham cua f(x) =

xl

n X ln(l

n x )

Gidi

Ta C O

: (hi(lnx))' =

Ne

n f(x)dx =

xl

nx

dx (•[In(lnx)]'

xlnx

hi(

ln x) hi(ln

x)

= In(lnx) +

C

40 I

Tim

ho nguye

n ham cua f(x) = (x +

1)

xQ + xe"

)

Gidi

Ta

CO :

(1

+ xe")' = e^

C x +

1)

(x + Ddx x = f (x)d

xd + xe'') e^Cx + Ddx

xe^Cl + xe") dCl +

xe")

[1

+ xe

^ IJ

-U +

xe"]

[(l + xe'')-(l + xe''-Dld

Cl + xe") rd(l + xe^ -

18

44] Ch

o hkm so f(x) = 3x^

m cu

a f(x) (x -

CO

: x

^ 3x + 2 = (

x

l)^(

x + 2)

B + Ox^

+ (

A +

B 2C)x + (2A

- 2

B + C)

|

X

-

ll + Inl

ham cua f(x) =

X (l +

2

fix) = —

X^

X (l +

x2)

X^

X l +

+1

fdx fd

x 1 d(x2

+1)

= J_ -Inl

xl + iln(x

2 +

46 I

Tim hp nguyen

ham cua f(x) = —

20

u

Gidi

Ta CO : f (x)dx = x^ + 4x^ + 4 x ^ + 1 dx x ^ - x

X J

x'^ + — + 4 x^

Gidi

Ta

CO :

f(x)dx = xdx

x^

+3x^

+2 [(x^ +

2) (x^

+ 1)

(x^

+ IXx

^ + 2) xdx

2 2

J x^

2 +2) +

C = -

m h

o nguye

n ham cua f(x) = 2-3x'

Gidi

Ta

CO :

f(x)dx =

dx

r d

x

3x

+ c

50

i Ti

m h

o nguye

n ham cua fix) =

-Gidi

Ta

CO :

"f(x)dx = (x* -

l)dx

f [(x

^ 5x + 1) - (x^ - 5x)](x'' -

Ddx

(x^ 5x)(x^

1 f

d(x^

- 5x) 1 fd(x^

-5x

+ D

x^ 5x

+ 1 x -5 x" 5 J + D 5x (x^ - x -5 x''

= i In I x*^

- 5x I

- i In I x^- 5

x +

11

+ C = -

-+ C

51

I Tim ho nguyen

ham cua f(x) =

x = rsm

X + s in

x co

s x

2 + sin

X

Ta

CO

: f(x ) =

Gidi

cos x + si n

X

cos x _ 2 cos x + si

n x cos x

- co

s x

2 + sin x

2 + sin x

cos x(

2 + sin x ) - co

s x cos x

F(x) =

2 + sin

X

-2 + sin

X

dx = X sin 2 +

= sin

x ln(2 + sinx) +

Gidi

sin3x.sin4x _ sinSx sin4

x _ sinSx sin4

-= — [sin5x +

sinSx sin9x +

Vay F(x)

= f (x)dx =

x sin 9

-x + sin x)d

x

- — cos 5

x — co

-s 3

x + — co

x + cos'*x)(sin®x +

F(x) =

dx

cos

X s in

n x

3 sin

^ x sin x(

l - si

n x )

+ C

61 Tim

X s + co X sm ) = a) f(x

X

b) g(x ) = -cos2

x

sin x + co s

X

DH Ngoai thuang -

1999

a) F(x

) =

b) G(x ) = sin

X

- co s

sin

X

+ co s

X

r-d(sin X

J si n

X

+ co s

X

= In sinx + cosx |

J

sin x + co

s x

G(x) sin

x + cosx +

ho nguye

n ha

m cii

a g(x ) = sinx.sin2x.cos5x

DH Bach khoa Ha

n 2x[si

n 6

x sin 4x ] =

- [sin2xsin6

x

= — [cos4x -

cos8x + cos6x -

Vay G(x ) =

— (co

s 4

x cos 8

-x + cos 6

x cos 2x ) d

— si

n 4

x — si

-n 8

x + — si

n 6

x — si

X

1 + sin 2

x

DH Bach khoa Ha

Noi

Ta

C O

: f(x ) = sin

x + cosx)^ o^^^sf

^ ^ - S X CO Z

1

4

Dat t=

x-— o

x = t + —

o d

x =

26

dt (t - l)(3t - 5)

A B

(t - l)(3t - 5) t - 1 3t - 5 1 = A(3t - 5) + B(t - 1)

c= 1 = (3

A + B)t - 5A

- B 3A

+ B = 0

-5A

- B = 1

-

—^

= il n|

- l|

+ i ln

: f(x ) =

x(cos X

- si

n x )

cos x

Vay F(x ) =

cos^ x(co

s X

sin x ) cos

n 1

1 tanx | + C

67 I Tim

ho nguye

n ha

m cu

a fix ) = cos^ x

: f(x ) =

CO S^

X

Gidi

1 + cos 2

x + —

= > d

t =

1 + cos 2

Nen f(x) = t-

2 sin

x

4(

cosx-l) 8

- — In

I cos

x 1

-1 + — I

m h

o nguye

n ham cua f(x) = sin^ x

3 si

n 4

x sin 6x -

3 si

n 2

x

BH SU pham Hd

3(sin4x sin2x) -

sin6x

= 6sinxcos3

x 2sin3x.cos3x =

2cos3x(3sinx sin3x)

-Va ta

CO

sin'^x =

f(x) =

3 sin

4 sin

^ x 8cos3x

8cos^3x

F(x) ^ f(x)dx

=

-co s

3xdx

co s

3x

1

8 J -d(sin 3x) f3

ho nguye

n ham ciia f(x) = hSng so)

DH Xdy dung -

1999

Gidi

^ , „ ,

s in

a CO

S x + si

n x c os

a si

n a co

s x s in a s in

Vay f(x)dx

- si

n a

cos^ x

sin xdx

cos^

X

•d(cos x)

X

sm

a In sin

Gidi

Ta

CO

: f(x) =

- (cos3

x + cosx)sin4x =

- sin4xcos3

x + — sin4xcos

= — (sin7x +

sinx + sin5x +

30

sin x(sin x - x cos x) + cot X + C

DH Bach khoa Ha Noi - 2000

x =

In I sinx I

- 2sin^

x + C

BHNgoqi thiCang -AI2000 3sinx 2 xsinx - Q g , X 4g cos Gidi sm x \ _ 3cosx) „ , „ o Q s x - ) = (4co O : f(x Ta C

Vay f(x)d

x = 4cos^xsin xdx

-3 sin xd

n

71 X

4j

(2 + 2sin2x)

DH Quoc gia Ha

^l

4j d

x + cos

1 — si

3x + si

DH Y TP.HCM -

x + 3sin8xcosx )

32

dx + sin

X

2

tan x.d(ta

n x ) + (sin x + co

s x ) dx

sinx +

C

82 Tim

b) g(x ) = cos^xcos2x

DH Ngoai thuang -

— (cosS

x + 3cosx)cos3x

3 „ n 2 — si x + n 4 — si + + - x (x)dx = Vay f

s 4x

Vay

g(x)dx = - 4

J

, 1 dx + —

2 cos 2

x + —

4 cos 4xd

x =

i X + —

sin 2

x + — si

n 4

x +

83 I Tim

ho nguye

n ha

m cii

a f(x ) = — sm

X

sin

X

+ co s

X

Gidi

Ta

CO :

X

+ co s

Va : cos xdx

sin X + cos X

sin xdx •(cos X - s i n x) sin X + cos x ( s i n x + cosx)

<=> asinx + bcosx = A(c.sinx + d.cosx) + B(c.cosx - d.sinx)

o asinx + bcosx = (Ac -Bd)sinx + (Ad + Bcjcosx

(tan X + l)dx + (cot x + l)dx = tanx - cutx + C

86 I T i m nguyen h a m F(x) ciia fix) = 2 sin 5x + Vx + - sao cho do t h i h a m so

5 f(x) va F(x) c^t nhau t a i m p t diem t r e n true tung

DH Quoc gia Hd Noi - B/1996

Gidi

CO : f(x ) = 2 si n5

x +

V

x + -

5

2 2 /—

3 Vx — x x + s 5 — co x) = F( =>

+ X

Do h

ai d

o t

hi cd

t n ha

u t

ai mo

t d ie

m tr en true tung, ne

n :

F(

0) = f(0)

+ C = - ^ C

87 Ti

m h

o n gu ye

n ha

m c ua f(x) =

e ™'"'sinx

Gidi

Ta

CO :

f(x)dx = |

e ế°'\sin

xdx =

,3 cos X

f

1 \

d(

3c os x) = -

n ha

m c ua f(x) = cos^

X

Gidi

Ta CO : (tanx) ' =

COS^

X

f(

x) dx

=

L^^-^=

fê^^^dCtanx) = ê"" + C

COS^

X

89

„, ,

- f/

X

s m

X + CO S

sm

X

- co s

X

Ta

CO :

f(x)dx =

Gidi

si n

X + CO S

X

>/sin

X cos X

-2

dx = (s in

X - co

s x ) 3d (s

X

- co s

x) 3

3 3/7

^ T „

= +

C =

- >/(s m

X

- co

s x

r + C

n ha

m c ua f(x) =

—

— n x si

36

t

Da

t Dat

j

* Da

t t = V

[du

= at

^dt

dv = sin

dt v = -

cos t

I = -2t^

cos t + 6

u = t^

Jdu = 2td

t

dv = cos td

t [v

= si

n t

I = -2t

^ co

s t + 6t

^ co

s t + 6t

^ si

n t + 12

dt

cos td

12sint

T I C H P H A N X A C D I N H

K I E N THLfC C d B A N

I T a C O cong thufc Newton - Leipnity

r b

f (x)dx = F(b) - F(a) vdi F(x) la nguyen h a m ciia f(x)

II Tinh chat

96 Tin

h I =

fi xd

x

0

(x^ + 1)^

Gidi

Da

t

t = x' + 1 =>

dt = 2xdx

Dd

i ca

dx r2

h I =

fi

dx

0

(x + I

f

DH Y duac TP.HCM

_dx_

^d(x.l) = lii

2

-2

I = -

I = -;; sin

x cos x , d n

-x,

- sin

x + cos

DH Da Ndng -

2000

Ta

CO

: I =

I

-d (sin

X

+ co

s x) ) s x + co X in (s -

4

Gidi

= -l

n sin

X

+ cos

X

Va

y I = -

ln

l + ln>

/2-ln2 2

Ti

m

A, B sao ch

o f(x) = AsinTtx +

B thoa man :

f '(1) =

Noi

B/1998

Gidi

Ta CO : f(x) = Asinrt

x +

B

f '(x ) = ATTCOSTIX

40

c:> (cos27; - cosO) + B(2 - 0) = 4 « 2B = 4 B = 2

DH Su pham Ki thudt - 1980 Gidi

1

0

x^ + 2x +

)

0

x^ + 2x +

9

+

-ln 2 x^ + 2x +

9

= i + i (l nl 2- ln 9)

f

1

x^ + 3x +

10

0 x^

+2x + 9

)

0

x^ + 2x +

9

= X

+

-ln 2 x^ + 2x + 9

= l.

il nl

1041 Chof(t ) =

4

0

Gidi

• t

f 4

0

(1

- CO S

2xf -

-dx

- 2 cos 2

x + cos^ 2

x + - (1 + cos 4x

dx

-c os 4x

= sin 4t -

0 o sin4t =

8sin2t

2sin2t.cos2t = 8sin2t

< =>

2sin2t(4 cos2t) =

0

sin 2t =

0

cos2t =

4 (v

6 nghi^m )

kit

o sin2

t =

0 c 2t =

kn

o t = —

-2 (104 _sin7ix)dx

BH Giao thong Van

tdi

1998

42

I O T I Tinh I = f ^ (e^" - n sin 7ix)dx

Gidi

Ta C O : I = r 1 (e^" - 7t sin 7Tx)dx =

r e 2 x + C O S 7:x

109 Tinh

I = (2

x co

s x)d

2x)

(1 + cos 2x) - - sin

- sin 2x)dx

3 „ cos2x

2

I—

ll o|

Tin

h da

o ham cua

am so F(x) = I

n ~ ^

c

f 1

- 1 — a : r Suy

Jo x

" +

1

Ta

C O

: F'(x) =

+ l

^

x^ >

-^x + 1 [(x^

+ 1) + V^xJ[(x2 + 1) - V2xl

2-x^

+1

x^ + V2x + 1

Do

do ix

"

-1

0 X^

+

1

dx =

2V2 F(x)

1 , 2-V

2 V2 In 2 + 2V2 '

m) Tin

h I = j

: I =

I =

dx

It

2-

2

- CO S

x)d

x 4

cos'^

xsin

^ x

tan

X

- cot

X

- sin

i=

V3-^

-(

i) = H:^ ^ 3 3 ^

i-44

0

= I M"JJ 8X

1

= 1 = 2J^ [£J^ Z-Zf^ +

8A-X SO

D + X

^O D-

= I

xp

XU TS -

TS

a

X

„U IS -

Z

li

X,

;0 D2

0

- 2 = (

2 u

i

8 ui )

-X =

Z^\

(x3

+ X)u

i • — -

X

=1

XP xS

I

xS + X

0

x3 xS + X

CO

: I =

x2 )

4-0

4-x

^

dx = -

4

= -4.-

In 4 x-

-2

2

dx

124| Cho f(x) = sin

ai so' A, B sao ch

o fix) =

A +

B ' cos

X

- sin

x y

b)

Tin

h I =

" 2

f (x)d

x 0

DH Su pham TP.HCM

- 1995

Gidi

a)

Tim A, B

Ta

CO

: f(x) = sin

X

= A + B

cos

X

+ sm

X

+ (

A B) si

-n x

sin

X

+ cos

l 2

b)

Tin

h I=

X

•dx =

t r

2 COS

X

- sin

X

cos

X

+ sin

X

dx

I =

— X I

n

2 2 cos

X

+ sin

X

7 T

4

125| Tin

h I = f

2

sin 3x

0

cos x +

1 dx

Ta

CO :

Gidi

sin 3

x 3 sin

X +

1

3 sin

(cosx

+1)

Vay 1 =

- si

n 3xd

x _ ^

sin xdx + 2

128 Tinh

I = f- 2cos-

X

Ta CO :

I =

= 2

Gidi

^ 2 cos^

X , „ f ^ cos

^ X.

cos xd

0

1 + sin

X

^2 f

2 co

s x- C'

x) =

2 si n

)

0

1 + sin

X

n 2

1

= 2 (1 ) =

129 Tinh

I = 2 sin^

xdx

BR Bach khoa TP.HCM

- BI2001

Gidi

Ta CO : I = 2 sin^

xdx = 2 (1 - cos^ x ) si

n xd

x

2 c,-

sin xdx +

2 cos^xd(co

s x ) = '^cos^ X

- co s

X 2

2

~

3

130 Tinh

I = 2 xdx

-1

x^

+ 2

DH Tong hap Ha

Noi

1993

Gidi

Ta CO : 2

J 2

n x^

+

2

-1

= (l n6 -l

-n 3 ) =

h bdi : f(x ) =

•

1

X neu X€

[0

;1

]

0 neu

> 0 thi f(x ) +oo

Vay f(x) khon

f

0 ne'u

X =

0

|l32| Xet si

r kh

a tic

h cu

a fix ) tre

n doa

n [1

; 0 ] vd

i fix ) = ^ -1 neu