ChM_teory2

N

>f xi 1 f xi @ , ɞɟ hi

Ihɬɪ f

¦i 1

hi

xi xi 1 .

2

Ɉɰɿɧɢɦɨ ɩɨɯɢɛɤɭ ɧɚ ɤɨɠɧɨɦɭ ɿɧɬɟɪɜɚɥɿ:

xi

h

>f xi 1 f xi @

h3

³ f x d x

i

i

§

Rhi Ii Ihi

f cc¨ xi 1

xi 1

2

6

©

Ɍɚɤɢɦ ɱɢɧɨɦ p

3 ɿ ɝɨɥɨɜɧɢɣ ɱɥɟɧ ɩɨɯɢɛɤɢ:

·

§ I

hi

I

hi

¨

¸

R0

©

2

¹

.

h

i

7

2

ɍɦɨɜɚ ɩɪɢɩɢɧɟɧɧɹ ɞɿɥɟɧɧɹ ɧɚɜɩɿɥ ɩɪɨɦɿɠɤɭ [xi 1, xi ]:

0

d

hi

.

Rhi

b

a

2

ɐɟ ɡɚɛɟɡɩɟɱɭɽ ɬɨɱɧɿɫɬɶ ɧɚ ɜɫɶɨɦɭ ɿɧɬɟɪɜɚɥɿ

N

N

hi

b a

Rh2

¦Rhi

d ¦

H

H .

i 1

2

i

1 b

a

b a

·	O hi		.
¸
2 ¹		5

ɓɟ ɨɞɧɟ ɡɚɫɬɨɫɭɜɚɧɧɹ ɩɪɢɧɰɢɩɭ Ɋɭɧɝɟ – ɜɢɫɨɤɨɬɨɱɧɟ ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɭ ɜɿɞ ɞɨɫɬɚɬɧɶɨ ɝɥɚɞɤɨʀ ɮɭɧɤɰɿʀ ɡɚ ɞɨɩɨɦɨɝɨɸ ɬɚɛɥɢɰɿ Ɋɨɦɛɟɪɝɚ. Ⱦɥɹ

ɩɨɛɭɞɨɜɢ ɰɿɽʀ ɬɚɛɥɢɰɿ ɨɛɱɢɫɥɢɦɨ ɡɚ ɞɨɩɨɦɨɝɢ ɫɤɥɚɞɟɧɨʀ ɤɜɚɞɪɚɬɭɪɧɨʀ

ɮɨɪɦɭɥɢ	ɬɪɚɩɟɰɿɣ		ɿɡ	ɫɬɚɥɢɦ ɤɪɨɤɨɦ		h ɩɨɫɥɿɞɨɜɧɿɫɬɶ ɡɧɚɱɟɧɶ
Ih Ih(0) ,Ih	Ih(0) ,Ih		Ih(0) ,Ih		Ih(0) ,..., ɹɤɿ	ɦɚɸɬɶ ɩɨɯɢɛɤɭ O(h2 ). Ɂɚ
2	2	4	4	8	8

ɞɨɩɨɦɨɝɨɸ ɟɤɫɬɪɚɩɨɥɹɰɿʀ Ɋɢɱɚɪɞɫɨɧɚ (3) ɡ ɤɨɟɮɿɰɿɽɧɬɚɦɢ ɥɿɧɿɣɧɨʀ ɤɨɦɛɿɧɚɰɿʀ

§

4

1

·

¨

,

¸

ɭɬɨɱɧɢɦɨ

ɰɿ

ɡɧɚɱɟɧɧɹ

(ɞɢɜ. ɬɚɤɨɠ ɮɨɪɦɭɥɭ

(4)). Ɉɬɪɢɦɚɽɦɨ

©

3

3

¹

ȼɨɧɢ

ɦɚɸɬɶ

ɩɨɯɢɛɤɭ O(h4 ) .

Ɂɧɨɜɭ

ɜɢɤɨɪɢɫɬɨɜɭɽɦɨ

Ih(1) ,Ih(1) ,Ih(1) ,... .

2

4

8

§

1 ·

16

ɟɤɫɬɪɚɩɨɥɹɰɿɸ Ɋɢɱɚɪɞɫɨɧɚ ɡ ɤɨɟɮɿɰɿɽɧɬɚɦɢ ɥɿɧɿɣɧɨʀ ɤɨɦɛɿɧɚɰɿʀ ¨

,

¸ .

©

15

15¹

Ɉɬɪɢɦɚɽɦɨ Ih(2) , Ih(2) ,... , ɹɤɿ ɦɚɸɬɶ ɬɨɱɧɿɫɬɶ O(h6 )

ɿ ɬ.ɞ.. Ɉɬɪɢɦɚɧɿ ɡɧɚɱɟɧɧɹ

48

ɦɨɠɧɚ ɪɨɡɦɿɫɬɢɬɢ ɜ ɬɚɤɿɣ ɬɚɛɥɢɰɿ Ɋɨɦɛɟɪɝɚ:

Ih(0)

I (0)	I (1)
h	h
2	2
I (0)	I (1)	I (2)
h	h	h
4	4	4
I (0)	I (1)	I (2)
h	h	h
8	8	8

Ih(3) 8

ȼɫɿ ɡɧɚɱɟɧɧɹ ɤɪɿɦ ɨɫɬɚɧɧɶɨɝɨ Ih(3) ɦɨɠɧɚ ɨɰɿɧɢɬɢ ɡɚ ɩɪɢɧɰɢɩɨɦ Ɋɭɧɝɟ (ɞɢɜ.

8

ɮɨɪɦɭɥɭ (2) ). ȼɢɤɨɪɢɫɬɚɧɧɹ ɮɨɪɦɭɥɢ (5) ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ Ih(02) k ɬɚ ɥɿɧɿɣɧɿ

91

ɤɨɦɛɿɧɚɰɿʀ (2) ɞɚɸɬɶ ɩɪɨɫɬɢɣ ɬɚ ɟɤɨɧɨɦɿɱɧɢɣ ɚɥɝɨɪɢɬɦ ɨɛɱɢɫɥɟɧɧɹ I .

ɉɨɱɚɬɤɨɜɟ ɡɧɚɱɟɧɧɹ h ɦɨɠɧɚ ɛɪɚɬɢ ɪɿɜɧɢɦ b a , ɚɛɨ b a , ɞɟ n ɰɿɥɟ. n

9.6. Ʉɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ ɧɚɣɜɢɳɨɝɨ ɚɥɝɟɛɪɚʀɱɧɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ [ɅɆɋ,
89-95], [ɋȽ, 180-183], [ȻɀɄ, 113-115, 102-108]
Ɋɨɡɝɥɹɧɟɦɨ ɿɧɬɟɝɪɚɥ	³b U x f x dx ,
I f				(1)
ɞɟ	a
U x 0 x >a,b@			³b U x xi dx		f .
			a
Ɋɨɡɝɥɹɧɟɦɨ ɡɚɞɚɱɭ ɩɨɛɭɞɨɜɢ ɤɜɚɞɪɚɬɭɪɧɨʀ ɮɨɪɦɭɥɢ
n	f xk ,
In f ¦ck				(2)

k 1

ɹɤɚ ɩɪɢ ɡɚɞɚɧɨɦɭ ɩ ɛɭɥɚ ɛ ɬɨɱɧɨɸ ɞɥɹ ɚɥɝɟɛɪɚʀɱɧɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ ɦɨɠɥɢɜɨ ɛɿɥɶɲɨɝɨ ɫɬɟɩɟɧɹ. Ɍɚɤɿ ɤɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ ɿɫɧɭɸɬɶ, ɜɨɧɢ ɧɚɡɢɜɚɸɬɶɫɹ

ɤɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ ɧɚɣɜɢɳɨɝɨ ɚɥɝɟɛɪɚʀɱɧɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ ɚɛɨ ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ (ɚɛɨ Ƚɚɭɫɫɚ – Ʉɪɢɫɬɨɮɟɥɹ).

ȼ (2) ɧɟɜɿɞɨɦɢɦɢ ɽ ck , xk , k				1,n	. Ȳɯ ɨɛɢɪɚɸɬɶ ɡ ɭɦɨɜɢ, ɳɨ (2) ɬɨɱɧɚ
ɞɥɹ ɛɭɞɶ-ɹɤɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ ɫɬɟɩɟɧɹ					p	, ɚ ɰɟ ɟɤɜɿɜɚɥɟɧɬɧɨ			ɭɦɨɜɿ, ɳɨɛ
ɮɨɪɦɭɥɚ ɛɭɥɚ ɬɨɱɧɨɸ ɞɥɹ ɮɭɧɤɰɿʀ				f	x	xD ,D	0,1,..., p .Ɂɜɿɞɫɢ ɨɬɪɢɦɭɽɦɨ
ɭɦɨɜɢ:
In xD	b			n
	³	U x xD dx	¦ck xkD , a
							0, p .		(3)
			k 1
	a			m o max . ɓɨɛ ɤɿɥɶɤɿɫɬɶ ɪɿɜɧɹɧɶ ɛɭɥɚ
Ɇɢ ɯɨɱɟɦɨ ɨɬɪɢɦɚɬɢ ɮɨɪɦɭɥɢ ɞɥɹ
ɪɿɜɧɨɸ ɤɿɥɶɤɨɫɬɿ ɧɟɜɿɞɨɦɢɯ ɧɚɦ ɩɨɬɪɿɛɧɨ, ɳɨɛ p							1 2n.

Ɂɚɞɚɱɚ 34 ɉɨɛɭɞɭɜɚɬɢ ɤɜɚɞɪɚɬɭɪɧɭ ɮɨɪɦɭɥɭ ɧɚɣɜɢɳɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ ( ɪɨɡɜ’ɹɡɚɬɢ ɫɢɫɬɟɦɭ ɪɿɜɧɹɧɶ (3)) ɞɥɹ a 1,b 1, x 1.

Ɍɟɨɪɟɦɚ Ƚɚɭɫɫɚ Ʉɜɚɞɪɚɬɭɪɧɚ ɮɨɪɦɭɥɚ (2) ɛɭɞɟ ɬɨɱɧɨɸ ɞɥɹ ɛɭɞɶ-ɹɤɨɝɨ
ɛɚɝɚɬɨɱɥɟɧɚ ɫɬɟɩɟɧɹ p		2n	1,ɬɨɛɬɨ		f	x		2n 1 ɬɨɞɿ ɿ ɬɿɥɶɤɢ ɬɨɞɿ,			ɤɨɥɢ
ɜɢɤɨɧɭɸɬɶɫɹ ɭɦɨɜɢ:		x	x1 x x2	... x			xn				x
1) ɩɨɥɿɧɨɦ	x							ɨɪɬɨɝɨɧɚɥɶɧɢɣ ɡ ɜɚɝɨɸ
ɞɨ ɛɭɞɶ-ɹɤɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ ɫɬɟɩɟɧɹ ɦɟɧɲɟ ɩ Qn 1 :
			³b Z x Qn 1 x U x dx					0;			(4)
			a
2) ɮɨɪɦɭɥɚ (2) ɽ ɤɜɚɞɪɚɬɭɪɧɨɸ ɮɨɪɦɭɥɨɸ ɿɧɬɟɪɩɨɥɹɰɿɣɧɨɝɨ ɬɢɩɭ ,
ɬɨɛɬɨ ɤɨɟɮɿɰɿɽɧɬɢ ɨɛɱɢɫɥɸɸɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ
							Z
			ck ³b U x			Z x				dx	(5)
								xk
			a		x	xk	c

92

ɇɟɨɛɯɿɞɧɿɫɬɶ. ɇɟɯɚɣ ɮɨɪɦɭɥɚ

(2)

ɬɨɱɧɚ

ɞɥɹ

ɛɚɝɚɬɨɱɥɟɧɚ

ɫɬɟɩɟɧɹ

p 2n 1, ɬɨɛɬɨ

I

f

In

f

f

x

2n

1 . Ɍɨɞɿ

I f

b

U x Z x Qn 1

x dx

n

0.

³

¦ck Z xk Qn 1 xk

k

1

a

Ɍɨɛɬɨ ɜɢɤɨɧɭɽɬɶɫɹ (4). Ɍɟɩɟɪ ɩɨɤɥɚɞɟɦɨ

f

x

x

n 1 2n 1 .

x

xj

c xj

Z

Ɉɬɪɢɦɚɽɦɨ

Z x

Z xk

b

b

n

n

³U x f x dx

³U x

dx

¦ck

¦ck Gkj

cj ,

xj

xk

c xj

a

a

x xj

c

k 1

xj

k 1

Z

Z

ɬɨɛɬɨ ɜɢɤɨɧɭɽɬɶɫɹ ɿ ɭɦɨɜɚ (5).

f x

Ⱦɨɫɬɚɬɧɿɫɬɶ. ɇɟɯɚɣ ɜɢɤɨɧɭɽɬɶɫɹ (4) ɿ (5). ɉɨɞɚɦɨ

2n 1

ɭ ɜɢɝɥɹɞɿ

Ɋɨɡɝɥɹɧɟɦɨ

f

x

x

Qn 1 x

Rn 1 x .

I( f ) ³b U x f x dx ³b U x Z x Qn 1 x Rn 1 x dx

a

Qn 1 xk

a

.

n

n

¦ck Z xk

¦ck Rn 1 xk

Ɍɚɤ ɹɤ Rn 1 xk

k 1

xk

Z xk Qn

k

1

xk , ɬɨ

f

1

xk

f

n

I( f )

¦ck f xk In ( f ).

k 1

Ɍɨɛɬɨ ɮɨɪɦɭɥɚ (2) ɽ ɬɨɱɧɨɸ ɞɥɹ ɛɭɞɶ-ɹɤɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ ɫɬɟɩɟɧɹ 2n

1.

Ɉɬɠɟ, ɡ ɬɨɱɧɿɫɬɸ ɞɨ ɫɬɚɥɨɝɨ ɦɧɨɠɧɢɤɚ ɛɚɝɚɬɨɱɥɟɧɢ

Z x ɫɩɿɜɩɚɞɚɸɬɶ ɡ

ɛɚɝɚɬɨɱɥɟɧɚɦɢ ɩ-ɬɨɝɨ ɫɬɟɩɟɧɹ ɨɪɬɨɝɨɧɚɥɶɧɨʀ ɫɢɫɬɟɦɢ ɛɚɝɚɬɨɱɥɟɧɿɜ. ɐɹ

ɫɢɫɬɟɦɚ ɨɪɬɨɝɨɧɚɥɶɧɚ ɧɚ ɩɪɨɦɿɠɤɭ

a,b@ ɡ ɜɚɝɨɸ

x .

ȼɢɜɱɢɦɨ ɞɟɹɤɿ ɜɥɚɫɬɢɜɨɫɬɿ ɤɜɚɞɪɚɬɭɪɧɢɯ ɮɨɪɦɭɥ Ƚɚɭɫɫɚ.

1) ɉɨɤɚɠɟɦɨ, ɳɨ ck , xk ɜɢɡɧɚɱɚɸɬɶɫɹ ɨɞɧɨɡɧɚɱɧɨ.

xn .

ɉɪɟɞɫɬɚɜɢɦɨ ɛɚɝɚɬɨɱɥɟɧ

Z x ɭ

ɜɢɝɥɹɞɿ

Z x

a0 a1 x ... an 1 xn 1

ɍɦɨɜɢ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɿ (4) ɩɪɢɣɦɭɬɶ ɜɢɝɥɹɞ

³b

U x Z x xD dx

³b

U x a0 a1x

... an 1xn 1 xn xD dx 0, D

0,n 1

a

a

ɚɛɨ

³b U x a0

a1x ... an

1xn 1 xD dx

³b U x xn xD dx

a

a

ɉɨɤɚɠɟɦɨ, ɳɨ ɜɿɞɩɨɜɿɞɧɚ ɨɞɧɨɪɿɞɧɚ ɫɢɫɬɟɦɚ ɪɿɜɧɹɧɶ

³b U x a0 a1x

... an 1xn 1

xD dx

0, D

0,n 1

(6)

a

ɦɚɽ ɽɞɢɧɢɣ ɪɨɡɜ’ɹɡɨɤ a0

a1 ...

an 1 0.

93

ɉɨɦɧɨɠɢɦɨ ɫɢɫɬɟɦɭ (6) ɧɚ aD

ɿ ɩɪɨɫɭɦɭɽɦɨ ɩɨ ɜɫɿɯ D

0,n 1:

n 1

b

a1x ... an 1xn 1 xD dx

¦aD ³U x a0

D 0

a

§ n 1

·

b

U x

n

1 n 1

aj aD x

D

x

j

dx

b

aj x

j

2

³

³

U x ¨

¦

¸

dx 0.

¦¦

¹

a

x

D

0 j 0

a

© j 0

Ɂɜɿɞɫɢ ɿ ɡ ɭɦɨɜɢ

0 ɜɢɩɥɢɜɚɽ, ɳɨ a0

a1

...

an

1

0. Ɍɨɦɭ ɿ ɜɿɞɩɨɜɿɞɧɚ

ɧɟɨɞɧɨɪɿɞɧɚ ɫɢɫɬɟɦɚ ɦɚɽ ɽɞɢɧɢɣ ɪɨɡɜ’ɹɡɨɤ. Ɉɬɠɟ ɿɫɧɭɽ ɽɞɢɧɢɣ ɛɚɝɚɬɨɱɥɟɧ Z x ɫɬɟɩɟɧɹ ɩ, ɹɤɢɣ ɨɪɬɨɝɨɧɚɥɶɧɢɣ ɡ ɜɚɝɨɸ x ɞɨ ɛɭɞɶ-ɹɤɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ

ɫɬɟɩɟɧɹ n 1.

2) ɉɨɤɚɠɟɦɨ, ɳɨ ɧɚɣɜɢɳɢɣ ɫɬɟɩɿɧɶ ɬɨɱɧɨɫɬɿ ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ p 2n 1. Ɂ t

ɬɟɨɪɟɦɢ ɜɢɩɥɢɜɚɽ, ɳɨ p 2n 1. ɉɨɤɚɠɟɦɨ, ɳɨ ɿɫɧɭɽ ɛɚɝɚɬɨɱɥɟɧ ɫɬɟɩɟɧɹ 2n ,

ɞɥɹ

ɹɤɨɝɨ

ɮɨɪɦɭɥɚ

ɧɟ

ɽ

ɬɨɱɧɨɸ.

Ⱦɥɹ

ɰɶɨɝɨ

ɜɜɟɞɟɦɨ

ɮɭɧɤɰɿɸ

f x

Z2 x 2n . Ɇɚɽɦɨ

I f

³b U x Z2 x dx

0,

ɚɥɟ

a

In f

¦ck Z2 xk

0.

n

Ɉɬɠɟ, I f

In f . Ɂɜɿɞɫɢ p

k 1

2n

1, ɬɨɛɬɨ p

2n

1.

3) Ʉɨɟɮɿɰɿɽɧɬɢ ɮɨɪɦɭɥ Ƚɚɭɫɫɚ ɞɨɞɚɬɧɿ,

ɬɨɛɬɨ

ck 0 . Ɋɨɡɝɥɹɧɟɦɨ

ɛɚɝɚɬɨɱɥɟɧɢ

ª

Z x

º2

M j

«

x xj

Z

»

,

«

c xj

»

ɹɤɿ ɦɚɸɬɶ ɫɬɟɩɿɧɶ 2n

¬

¼

2 ɿ ɜɥɚɫɬɢɜɨɫɬɿ:

³b

U x M j

x dx 0 .

Mi

xk

Gik , I(M j )

a

Ɍɚɤ ɹɤ ɞɥɹ ɰɢɯ ɛɚɝɚɬɨɱɥɟɧɿɜ ɫɩɪɚɜɟɞɥɢɜɚ ɬɟɨɪɟɦɚ Ƚɚɭɫɫɚ, ɬɨ

I M j In M j

¦ck M j xk

¦ck

ª

Z

x

º

¦ck G2jk

«

k

»

cj .

n

n

«

Z

»

2

n

k 1

k 1

¬

x xj

c

xj

¼

k 1

Ɂɜɿɞɫɢ ɜɢɩɥɢɜɚɽ, ɳɨ cj 0,

x

a,b@

f x C2n >a,b@. Ɍɨɞɿ

j

1,n.

4) Ɍɟɨɪɟɦɚ

ɇɟɯɚɣ ɜɚɝɨɜɚ ɮɭɧɤɰɿɹ

0

x

ɿɫɧɭɽ ɬɨɱɤɚ [ >a,b@

ɬɚɤɚ, ɳɨ

f

[

³b U x Z2 x dx.

Rn

f

2n

(7)

2n !

a

Ɋɨɡɝɥɹɧɟɦɨ ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ ɛɚɝɚɬɨɱɥɟɧ ȿɪɦɿɬɚ ɡ ɞɜɨɤɪɚɬɧɢɦɢ ɜɭɡɥɚɦɢ

, f c

H2cn 1 xi , i

.

H2n 1 x : f xi

H2n 1 xi

xi

1,m

94

Ⱦɥɹ ɧɶɨɝɨ

x H2n 1

x

2n !

Z

x .

r2n 1 x f

f

2n [

2

Ɂɜɿɞɫɢ

2n [

R2n 1 x ³b U x r2n 1 x dx

f

³b U x Z2 x dx.

a

2n ! a

9.7. ɑɚɫɬɢɧɧɿ ɜɢɩɚɞɤɢ ɤɜɚɞɪɚɬɭɪɧɨʀ ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ [ɅɆɋ, 99]

1) Ɋɨɡɝɥɹɧɟɦɨ ɜɿɞɪɿɡɨɤ >

1,1@ ɿ ɜɚɝɨɜɢɣ ɦɧɨɠɧɢɤ

x

1, ɬɨɛɬɨ ɜɢɜɟɞɟɦɨ

ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɭ

³1

f x

dx.

1

	ɓɨɛ ɡɧɚɣɬɢ ɜɭɡɥɢ ɤɜɚɞɪɚɬɭɪɧɚ ɮɨɪɦɭɥɢ ɪɨɡɝɥɹɧɟɦɨ ɛɚɝɚɬɨɱɥɟɧɢ
Ʌɟɠɚɧɞɪɚ n 1 Ln 1 x 2n 1 xLn x		nLn 1 x , L0 1, L1 x	x .
Ȼɚɝɚɬɨɱɥɟɧɢ Ʌɟɠɚɧɞɪɚ ɡɚɞɨɜɨɥɶɧɹɸɬɶ ɭɦɨɜɚɦ ɬɟɨɪɟɦɢ 1 (ɩɭɧɤɬ 1), ɬɨɦɭ
Z x	Ln x

ɿ ɜɭɡɥɚɦɢ ɤɜɚɞɪɚɬɭɪɧɨʀ ɮɨɪɦɭɥɢ ɽ ɤɨɪɟɧɿ ɰɶɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ. ȼɚɝɨɜɿ

ɦɧɨɠɧɢɤɢ ɰɿɽʀ ɮɨɪɦɭɥɢ ɨɛɱɢɫɥɸɸɬɶɫɹ ɡɚ ɮɨɪɦɭɥɨɸ

1

Z x

ck

³1

dx,

x

xk

c xk

Z

ɚ ɡɚɥɢɲɤɨɜɢɣ ɱɥɟɧ

Rn f

1

n! 2

2n

[ .

22n

2n

1 ! 2n ! 2n ! f

ɉɪɢɤɥɚɞ. ɉɨɛɭɞɭɽɦɨ ɤɜɚɞɪɚɬɭɪɧɭ ɮɨɪɦɭɥɭ

1

x dx

n

xk .

³ f

¦ck f

0

k 1

ɉɪɢ n 2

ɩɨɬɪɿɛɧɨ

ɡɧɚɣɬɢ

c0 ,c1, x0 , x1 .

Ɂɚɦɿɧɨɸ

t 2x 1 ɩɟɪɟɜɟɞɟɦɨ

x >0,1@ ɧɚ ɩɪɨɦɿɠɨɤ t

> 1,1@. Ɂɚɩɢɲɟɦɨ L2

t

3

t2

1

. Ɂɜɿɞɫɢ

2

2

L2 x

3 2x 1 1 12x2

12x 2

6x

2

6x 1 0 .

2

2

Ɂɜɿɞɫɢ x1

3 3 , x2

3

3 . Ɂɚ ɮɨɪɦɭɥɨɸ (5) ɡɧɚɣɞɟɦɨ

6

6

1

x

x

2

1

1

x

x

1

c

³0

dx

,

c

³0

1

dx

.

2

1

x1

x2

2

x2

x1

2

Ɍɨɛɬɨ ɤɜɚɞɪɚɬɭɪɧɚ ɮɨɪɦɭɥɚ ɦɚɽ ɜɢɝɥɹɞ

95

³ f x dx

1

§

§

3

3

·

§ 3

3 ··

2

¨

f ¨

6

¸ f ¨

6

¸¸ .

1

©

©

¹

©

¹¹

0

1,1@

U x

1

x2

2) Ɋɨɡɝɥɹɧɟɦɨ ɜɿɞɪɿɡɨɤ

ɿ

ɜɚɝɚ

1

, ɬɨɛɬɨ ɜɢɜɟɞɟɦɨ

2

ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɭ

2 .

I f

³

1

f

x dx

1

1

x

Ȼɚɝɚɬɨɱɥɟɧɢ ɑɟɛɢɲɨɜɚ ɡɚɞɨɜɨɥɶɧɹɸɬɶ ɭɦɨɜɚɦ ɬɟɨɪɟɦɢ 1 (ɩ.1), ɬɨɦɭ

Z x

Tn x

1

cos narccos x .

2

n

1

ȼɭɡɥɚɦɢ ɤɜɚɞɪɚɬɭɪɧɨʀ ɮɨɪɦɭɥɢ ɽ ɤɨɪɟɧɿ ɰɶɨɝɨ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ, ɬɨɛɬɨ

ɤɨɪɟɧɿ ɪɿɜɧɹɧɧɹ cos narccos x

0 . Ɂɜɿɞɫɢ

ɯk

2k 1

cos

, k 1,n .

2n

ȼɿɞɩɨɜɿɞɧɿ ɤɨɟɮɿɰɿɽɧɬɢ ɨɛɱɢɫɥɸɸɬɶɫɹ ɡɚ ɮɨɪɦɭɥɚɦɢ

³1 1

x2 Tnc xk x xk n

1

Tn x dx

ck

, k

1,n.

Ɉɬɠɟ, ɤɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ ɧɚɣɜɢɳɨɝɨ ɫɬɟɩɟɧɹ ɬɨɱɧɨɫɬɿ (ɰɿ ɮɨɪɦɭɥɢ ɧɚɡɢɜɚɸɬɶ ɮɨɪɦɭɥɚɦɢ ȿɪɦɿɬɚ) ɦɚɸɬɶ ɜɢɝɥɹɞ

n

In f

¦k 1

f xk

,

(1)

n

ɞɟ xk – ɤɨɪɟɧɿ ɛɚɝɚɬɨɱɥɟɧɚ ɑɟɛɢɲɨɜɚ.

Ɂɚɥɢɲɤɨɜɢɣ ɱɥɟɧ ɦɚɽ ɜɢɝɥɹɞ

f

22n 1 2n !

f 2n [ .

Rn

3) Ɋɨɡɝɥɹɧɟɦɨ ɩɪɨɦɿɠɨɤ

,

ɿ ɜɚɝɭ

U x e x2

, ɬɨɛɬɨ ɜɢɜɟɞɟɦɨ

ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɭ

f

f x dx.

³e x2

Ɂɚ ɬɟɨɪɿɽɸ

f

Z x

Hn x 1 n ex2

d

n

e x2 ,

2

ɞɟ Hn x ɛɚɝɚɬɨɱɥɟɧɢ ȿɪɦɿɬɚ.

dx

Ȼɚɝɚɬɨɱɥɟɧɢ ȿɪɦɿɬɚ ɨɛɱɢɫɥɸɸɬɶɫɹ ɬɚɤɨɠ ɡɚ

ɪɟɤɭɪɟɧɬɧɢɦɢ ɮɨɪɦɭɥɚɦɢ

Hn 1 x

2xHn x 2nHn 1 x , H 1

0, H0

1 .

Ʉɨɟɮɿɰɿɽɧɬɢ ɤɜɚɞɪɚɬɭɪɧɨʀ ɮɨɪɦɭɥɢ ɨɛɱɢɫɥɸɸɬɶɫɹ ɡɚ ɮɨɪɦɭɥɚɦɢ

ck

2n 1 n!

.

Hnc

xk

2

@

>

Ɂɚɥɢɲɤɨɜɢɣ ɱɥɟɧ

96

Rn f

n!

2n [ .

1 H1 x

2n 2n !

f

ɇɚɩɪɢɤɥɚɞ, ɩɪɢ n

2x. Ʉɨɪɿɧɶ x0

0,

f

c

0

e x2 dx

.

³f

0 .

Ʉɜɚɞɪɚɬɭɪɧɚ ɮɨɪɦɭɥɚ ɦɚɽ ɜɢɝɥɹɞ:

I1

x

f

4) Ɋɨɡɝɥɹɧɟɦɨ ɜɿɞɪɿɡɨɤ

0,

@

ɿ ɜɚɝɨɜɢɣ ɦɧɨɠɧɢɤ U x xDe x , ɬɨɛɬɨ

ɜɢɜɟɞɟɦɨ ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ ɞɥɹ ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɭ

f³ xDe x f

x dx.

Ɂɚ ɬɟɨɪɿɽɸ

0

xD ne x ,

Z x LDn x 1 n x Dex

d n

n

dx

ɞɟ LDn x ɛɚɝɚɬɨɱɥɟɧɢ Ʌɚɝɟɪɚ. Ʉɨɟɮɿɰɿɽɧɬɢ ɨɛɱɢɫɥɸɸɬɶɫɹ ɡɚ ɮɨɪɦɭɥɚɦɢ

c

P n

1 P n

1

.

k

xk >LDn xk

@2

Ɂɚɥɢɲɤɨɜɢɣ ɱɥɟɧ ɩɪɢ D 0 ɪɿɜɧɢɣ

n! 2

Rn f

2n

[ .

2n ! f

7) ȱɧɬɟɝɪɭɜɚɧɧɹ ɲɜɢɞɤɨ ɨɫɰɢɥɸɸɱɢɯ ɮɭɧɤɰɿɣ.

Ɋɨɡɝɥɹɧɟɦɨ ɿɧɬɟɝɪɚɥ

³b

f x e jZx dx, j2

I f

1.

ɉɪɢ ɜɟɥɢɤɢɯ

a

ɡɚɫɬɨɫɭɜɚɧɧɹ ɫɤɥɚɞɟɧɢɯ

ɤɜɚɞɪɚɬɭɪɧɢɯ ɮɨɪɦɭɥ

ɿɧɬɟɪɩɨɥɹɰɿɣɧɨɝɨ ɬɢɩɭ ɩɪɢɜɨɞɢɬɶ ɞɨ ɜɟɥɢɤɨʀ ɩɨɯɢɛɤɢ ɿ ɩɪɢ ɦɚɥɢɯ ɤɪɨɤɚɯ h .

Ɋɨɡɝɥɹɧɟɦɨ e jZx

ɹɤ ɜɚɝɨɜɢɣ ɤɨɟɮɿɰɿɽɧɬ, ɬɨɛɬɨ

U x

e jZx . Ɂɚɦɿɧɢɦɨ >a,b@

ɧɚ

> 1,1@: xi

a b

b

a

di , di > 1, 1@, i

(ɜɭɡɥɢ

ɦɨɠɭɬɶ

ɛɭɬɢ

ɧɟ

1,n

2

2

1 i

ɪɿɜɧɨɜɿɞɞɚɥɟɧɿ, ɹɤɳɨ ɪɿɜɧɨɜɿɞɞɚɥɟɧɿ, ɬɨ di

2

, i

1,n).

f x

n

ɡ

Ɂɚɦɿɧɢɦɨ

ɧɚ ɿɧɬɟɪɩɨɥɹɰɿɣɧɢɣ ɛɚɝɚɬɨɱɥɟɧ

Ʌɚɝɪɚɧɠɚ

Ln 1 x

ɜɭɡɥɚɦɢ xi

ɿ ɨɬɪɢɦɚɽɦɨ ɮɨɪɦɭɥɭ

Ln 1 x e jZx dx,

In f ³b

(2)

a

ɹɤɚ ɛɭɞɟ ɬɨɱɧɨɸ ɞɥɹ ɜɫɿɯ ɛɚɝɚɬɨɱɥɟɧɿɜ ɧɟ ɜɢɳɟ n 1 ɫɬɟɩɟɧɹ . Ɍɨɛɬɨ, ɹɤɳɨ ɜ

(2) ɩɿɞɫɬɚɜɢɬɢ ɛɚɝɚɬɨɱɥɟɧ Ʌɚɝɪɚɧɠɚ, ɬɨ ɦɨɠɧɚ ɨɛɱɢɫɥɢɬɢ ɿɧɬɟɝɪɚɥ ɿ ɨɬɪɢɦɚɬɢ

ɤɜɚɞɪɚɬɭɪɧɭ ɮɨɪɦɭɥɭ

97

­

½

n

§

·

1 §

n

·

Sn f

2

2

2

, Di

p ³¨

d

d

¸exp jp[ d[

exp® jZ

¾¦Di ¨Z

¸ f xi

Z

b a

¯

a b

¿ i 1

©

b a

¹

¨

[

dk

¸

1

©

k 1 i

k

¹

k i

.

ɉɪɢ n

3,d1

1,d2

0,d3

1 – ɰɟ ɮɨɪɦɭɥɚ Ɏɿɥɨɧɚ. Ɇɨɠɧɚ ɛɪɚɬɢ ɿ ɛɿɥɶɲɟ

ɬɨɱɨɤ,

ɧɚɩɪɢɤɥɚɞ n

5,d1

1,d2

1

,d

0,d4

1

,d5

1. ɐɿ ɮɨɪɦɭɥɢ ɧɟ

2

3

2

ɩɨɬɪɿɛɧɨ ɡɚɫɬɨɫɨɜɭɜɚɬɢ, ɤɨɥɢ ɧɟɦɚɽ ɲɜɢɞɤɨ ɨɫɰɢɥɸɸɱɨɝɨ ɦɧɨɠɧɢɤɚ.

9.8. Ɉɛɱɢɫɥɟɧɧɹ ɧɟɜɥɚɫɧɢɯ ɿɧɬɟɝɪɚɥɿɜ [ȻɀɄ, 146-153] Ɋɨɡɝɥɹɧɟɦɨ ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɿɜ ɡ ɬɚɤɢɦɢ ɨɫɨɛɥɢɜɨɫɬɹɦɢ :

ɚ) ɿɧɬɟɝɪɚɥɢ ɞɪɭɝɨɝɨ ɪɨɞɭ, ɬɨɛɬɨ			o ;
I	b	F x dx, F x
	³		xoa xob
	a

ɛ) ɿɧɬɟɝɪɚɥɢ ɩɟɪɲɨɝɨ ɪɨɞɭ			f³ F x dx.
		I
			a
Ɋɨɡɝɥɹɧɟɦɨ ɫɩɨɱɚɬɤɭ ɿɧɬɟɝɪɚɥɢ ɞɪɭɝɨɝɨ ɪɨɞɭ, ɬɨɛɬɨ
I	b	F x dx	F x	o	.
	³			xoa xob
	a

1) Ɇɭɥɶɬɢɩɥɿɤɚɬɢɜɧɢɣ ɫɩɨɫɿɛ. ɉɪɟɞɫɬɚɜɢɦɨ ɩɿɞɿɧɬɟɝɪɚɥɶɧɭ ɮɭɧɤɰɿɸ ɭ
ɜɢɝɥɹɞɿ	F x	x f x , ɩɪɢɱɨɦɭ	x	– ɨɫɨɛɥɢɜɚ, ɚ f x – ɝɥɚɞɤɚ. Ⱦɚɥɿ ɞɥɹ
ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɭ				³b U x f x dx
		I I~ f
				a
ɜɢɤɨɪɢɫɬɨɜɭɽɦɨ ɜɿɞɩɨɜɿɞɧɿ ɤɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ Ƚɚɭɫɫɚ.
ɉɪɢɤɥɚɞ 1 ɉɨɬɪɿɛɧɨ ɨɛɱɢɫɥɢɬɢ ɿɧɬɟɝɪɚɥ
		I	³1	dx	x	4 .
Ɍɨɱɤɢ x	1 ɽ ɨɫɨɛɥɢɜɢɦɢ.		1	1
ɉɪɟɞɫɬɚɜɢɦɨ ɩɿɞɿɧɬɟɝɪɚɥɶɧɭ ɮɭɧɤɰɿɸ ɭ ɜɢɝɥɹɞɿ:
		F x	1			1
			1	x2		1 x2 ,
ɨɬɪɢɦɚɽɦɨ ɿɧɬɟɝɪɚɥ ɜɢɝɥɹɞɭ			U x			f x
		1	1			dx
ɞɟ U x		I ³0 1 x2			1 x2 ,
		1
	1	x2 .

98

Ⱦɚɥɿ ɜɢɤɨɪɢɫɬɨɜɭɽɦɨ ɤɜɚɞɪɚɬɭɪɧɭ ɮɨɪɦɭɥɭ ȿɪɦɿɬɚ (1) ɡ ɩɨɩɟɪɟɞɧɶɨɝɨ

ɩɭɧɤɬɭ ɿ ɨɛɱɢɫɥɸɽɦɨ ɿɧɬɟɝɪɚɥ.

ɉɪɢɤɥɚɞ 2 Ɉɛɱɢɫɥɢɬɢ ɿɧɬɟɝɪɚɥ

³ln sin x dx.

I

Ɉɫɨɛɥɢɜɿ ɬɨɱɤɢ x 0, x

0

.

Ɂɜɟɞɟɦɨ ɰɸ ɨɫɨɛɥɢɜɿɫɬɶ ɞɨ ɫɬɟɩɟɧɟɜɨʀ:

U x

x 1

x ,

ɬɨɞɿ

xo o

f

x

x

x

ln sin x

0.

0,

Ⱦɥɹ ɡɧɚɯɨɞɠɟɧɧɹ ɿɧɬɟɝɪɚɥɭ ɡ ɬɚɤɢɦ

x ɡɚɫɬɨɫɨɜɭɽɦɨ ɤɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ

ɑɟɛɢɲɨɜɚ. ɇɟɩɪɢɽɦɧɨɫɬɿ

ɜɢɧɢɤɚɸɬɶ,

ɨɫɤɿɥɶɤɢ

f x

xo0, of

(ɯɨɱɚ

ɤɜɚɞɪɚɬɭɪɧɿ ɮɨɪɦɭɥɢ ɞɚɜɚɬɢɦɭɬɶ ɧɚɛɥɢɠɟɧɟ ɡɧɚɱɟɧɧɹ). Ɍɨɦɭ ɡɚɫɬɨɫɨɜɭɸɬɶ
ɞɪɭɝɢɣ ɫɩɨɫɿɛ ɪɨɡɜ’ɹɡɚɧɧɹ ɩɪɨɛɥɟɦɢ:
F x	f	x	\ x ,	ɉɪɟɞɫɬɚɜɢɦɨ ɩɿɞɿɧɬɟɝɪɚɥɶɧɭ ɮɭɧɤɰɿɸ ɭ ɜɢɝɥɹɞɿ
2)		ɞɢɬɢɜɧɢɣ ɫɩɨɫɿɛ.
ɩɪɢɱɨɦɭ		\ x –ɨɫɨɛɥɢɜɚ,		f x – ɝɥɚɞɤɚ. Ɋɨɡɛɢɜɚɽɦɨ ɿɧɬɟɝɪɚɥ ɧɚ ɞɜɚ:
I I1	I2 .		f x dx – ɨɛɱɢɫɥɸɸɬɶ ɱɢɫɟɥɶɧɨ (ɧɚɩɪɢɤɥɚɞ, ɮɨɪɦɭɥɢ ɋɿɦɩɫɨɧɚ ɱɢ
	I1	³b

a

ɬɪɚɩɟɰɿɣ

I ), ³b \ x dx – ɩɪɨɛɭɸɬɶ ɨɛɱɢɫɥɢɬɢ ɚɧɚɥɿɬɢɱɧɨ (ɦɨɠɥɢɜɨ ɚɩɪɨɤɫɢɦɭɜɚɬɢ2 a ɮɭɧɤɰɿɸ \ x , ɧɚɩɪɢɤɥɚɞ, ɪɹɞɨɦ).

ɉɪɢɤɥɚɞ 3 Ɉɛɱɢɫɥɢɬɢ ɿɧɬɟɝɪɚɥ ɡ ɩɪɢɤɥɚɞɭ 2:
	I	³ln sin x dx.
		0
	I	2 ³2ln sin x dx.

o

ɉɪɟɞɫɬɚɜɢɦɨ

ln sin x ln sin x ln x.

Ɉɬɪɢɦɚɽɦɨ ɞɜɚ ɿɧɬɟɝɪɚɥɢ:											x
	b		sin x
I1	³ln			dx ɨɛɱɢɫɥɸɽɦɨ ɱɢɫɟɥɶɧɨ,
			x
	a						§				·
		2				2
ɚ I2		0						©			¹
		³ln xdx xln x x				0	2 ¨ln		2		1¸ .

99

Ɋɨɡɝɥɹɧɟɦɨ ɬɟɩɟɪ ɿɧɬɟɝɪɚɥɢ ɩɟɪɲɨɝɨ ɪɨɞɭ

I

f³ F

x dx

a

ɇɟɯɚɣ ɚ>0 .Ɂɪɨɛɢɦɨ ɡɚɦɿɧɭ t

x

a

,

x

a

. Ɍɨɞɿ

1

x

§

t

1

a

·

dt

I

a

F

¨

¸

t

2 ,

ɚ ɰɟ ɿɧɬɟɝɪɚɥ ɞɪɭɝɨɝɨ ɪɨɞɭ.

³0

©1

t ¹ 1

əɤɳɨ ɚ=0, ɬɨ ɪɨɛɢɦɨ ɡɚɦɿɧɭ t

e

x , x

lnt , ɬɨɞɿ

1

dt

³ F lnt

I

.

t

0

Ɂɧɨɜɭ ɨɬɪɢɦɭɽɦɨ ɿɧɬɟɝɪɚɥ ɞɪɭɝɨɝɨ ɪɨɞɭ.

əɤɳɨ ɚ<0 (ɧɟ ɦɨɠɧɚ ɡɪɨɛɢɬɢ ɡɚɦɿɧɭ t x a , ɬɨɦɭ ɳɨ ɜɢɧɢɤɚɽ x

ɨɫɨɛɥɢɜɿɫɬɶ ɜ ɬɨɱɰɿ x 0 ), ɪɨɡɛɢɜɚɽɦɨ ɿɧɬɟɝɪɚɥ ɧɚ ɞɜɚ:
0	f

I ³ F x dx ³ F x dx

ao

ɿɨɛɱɢɫɥɸɽɦɨ ɡɚ ɞɨɩɨɦɨɝɨɸ ɩɨɩɟɪɟɞɧɿɯ ɩɭɧɤɬɿɜ.

Ɇɭɥɶɬɢɩɥɿɤɚɬɢɜɧɢɣ				ɫɩɨɫɿɛ		ɨɛɱɢɫɥɟɧɧɹ ɿɧɬɟɝɪɚɥɿɜ				ɩɟɪɲɨɝɨ	ɪɨɞɭ
				F x		x f		x ,
ʉɪɭɧɬɭɽɬɶɫɹ ɧɚ ɩɪɟɞɫɬɚɜɥɟɧɿ ɩɿɞɿɧɬɟɝɪɚɥɶɧɨʀ ɮɭɧɤɰɿʀ ɭ ɜɢɝɥɹɞɿ
ɞɟ, ɧɚɩɪɢɤɥɚɞ,				U x	xD e x , x [0,f).
Ɍɚɤɢɣ	ɜɚɝɨɜɢɣ	ɤɨɟɮɿɰɿɽɧɬ			ɜɿɞɩɨɜɿɞɚɽ				ɛɚɝɚɬɨɱɥɟɧɚɦ	Ʌɚɝɟɪɚ.	ɉɪɢ
x	f,f , U x	e x2 ɩɪɢɯɨɞɢɦɨ ɞɨ ɛɚɝɚɬɨɱɥɟɧɿɜ ȿɪɦɿɬɚ.
ɓɟ ɨɞɢɧ ɫɩɨɫɿɛ ʉɪɭɧɬɭɽɬɶɫɹ ɧɚ ɨɛɪɿɡɚɧɧɿ ɜɟɪɯɧɶɨʀ ɝɪɚɧɢɰɿ. Ⱦɥɹ ɰɶɨɝɨ
ɿɧɬɟɝɪɚɥ ɡɚɩɢɲɟɦɨ ɭ ɜɢɝɥɹɞɿ
		f				b			³ F x dx .
		I	³	F x dx		³ F x dx
			a			a			b
ȼɟɪɯɧɹ ɝɪɚɧɢɰɹ b ɨɛɱɢɫɥɸɸɬɶ ɡ ɭɦɨɜɢ								H
					f	x dx
					³F				,
								2
					b
ɞɟ – ɡɚɞɚɧɚ ɬɨɱɧɿɫɬɶ. Ⱦɥɹ ɨɛɱɢɫɥɟɧɧɹ ³b F x dx ɜɢɤɨɪɢɫɬɨɜɭɸɬɶ ɤɜɚɞɪɚɬɭɪɧɿ
ɮɨɪɦɭɥɢ ɫɤɥɚɞɟɧɨɝɨ ɬɢɩɭ.						a

9.9. Ɉɛɱɢɫɥɟɧɧɹ ɤɪɚɬɧɢɯ ɿɧɬɟɝɪɚɥɿɜ [ȼɨɥɤɨɜ, 125-129], [Ʉɚɥɢɬɤɢɧ, 108-113,

121-123]

Ɋɨɡɝɥɹɧɟɦɨ ɿɧɬɟɝɪɚɥ

100