matan-1_2

0 3 + 6$ $ ( '

7 D

t t

0 ≤ F (t) =	f (x)dx ≤ g(x)dx = G(t).
a	a

" I2 D : ! E <

G(t) , : : , F (t) ! ! E I1 D : , 7

b b

0 ≤ a

f

x

dx

lim F t

lim G(t) =

g(x)dx.

(

)

= t→b− ( )

≤ t→b−

a

" I1 D :

+

= lim F (t)

lim G(t).

∞

t

b

−

≤ t

b

−

→

→

#"L " 1 2 N 7 O

I1 = a

b

I2 = a

b

f (x)dx,

g(x)dx

! ! b

(a, b) 0 ≤ f (x) < g(x)# "

'

lim	f (x)	= A > 0.
x→b− g(x)
) I1 I2 #
!
ε > 0 (ε < A) c [a, b) A − ε <			f (x)		< A + ε, x (c, b),
			g(x)
g(x) > 0 [a, b):
(A − ε)g(x) < f (x) < (A + ε)g(x), x (c, b).						(7.1)
$ D 7 I2 D 7

b

g(x)dx

c

0 4 ) +$ ? ( ( (

N -O: D 7

b

(A + ε)g(x)dx

c

N 3O C M 7 7 1 3 D 7

b

f (x)dx,

c

: I1 N -O

$: 7 : , D 7 I2 D 7 I1 L , 7

N1-O

/ 8 1 A * ? ? 9 ?

(" =- g C1[a, +∞) f C[a, +∞) F (x) 0

f (x)# . '

+∞

gxF (x)dx = A,

a

'

lim g(x)F (x) = B,

x→+∞

'

+∞

f (x)g(x)dx = B − g(a)F (a) − A.

a

C A R: A > a 7 7

A	A
	f (x)g(x)dx = g(A)F (A) − g(a)F (a) −	F (x)g (x)dx.

a

a

0 4 ) +$ ? ( ( (

7 , <

7 C 7 7 ' < ( <

7 f 7

g C D A → +∞: , 7 <

7

) 7 7: , 77 =- 7 7 <

, D + 7 <

< 5 D

#"L " =- g C1[a, +∞) f C[a, +∞) F (x) 0

f (x) !'

2 |F (x)| ≤ M x > a3

x→+∞	(	) = 0		|g | =	+∞		)|	∞
					a	| (
lim	g x		, I			g x	dx <		.

) '

+∞

f (x)g(x)dx.

a

5 7 77

=- 5 : |g (x)F (x)| ≤ M|g (x)|: 7

+∞

g (x)F (x)dx D D 7

a

I|g | % 7 :

|g(x)F (x)| ≤ M|g(x)| → 0 x → +∞,

7

lim g(x)F (x) = 0.

x→+∞

#"L " = 3 g C1[a, +∞) f C[a, +∞) F (x) 0

f (x) !'

2 |F (x)| ≤ M x > a3 g

x > a

lim g(x) = 0.

x→+∞

0 4 ) +$ ? ( ( (

) '

+∞

f (x)g(x)dx.

a

# 7 : 7 : ,

A	|	(			A		(
A→+∞ a			x	)| = − A→+∞ a		g		)	dx	=
lim	g			dx	lim			x

= lim (g(a) − g(A)) = g(a)

A→+∞

7 <

C$" =- $

+∞	xα dx,	+∞	xα dx
I1 = a		I2 = a
	sin x		cos x

α > 0: a > 0 7 7 = 3 <

7 7 I1 NI2 7 , O C <

F (x) = − cos x f (x) = sin x , :

g(x) = x−α 7 (a, +∞) 7

7 x → +∞ : I1 D <

# 7 D , D

5 M 7 7

	b
	I = a	f (x)dx,
7	, b C
a = b0 < b1	< · · · < bk	< · · · < b,	klim bk = b.
			→∞

# 7 ,

∞bk+1

S = f (x)dx.

k=0 bk

0 4 ) +$ ? ( ( (

#"L " = 2 . I S

I = S

n	bk+1		bn+1	b
n→∞ k=0		n→∞
bk		f (x)dx = lim	a	a
lim			f (x)dx =	f (x)dx.

#"L " = 4 . f (x) ≥ 0 x [a, b) S

I I = S

! 7 7 b (a, b) n (bn (b , b) C M 7 : ,

f : , 7

b	bn	n−1	bk+1
F (b ) =	f (x)dx ≤	f (x)dx = k=0		f (x)dx ≤ I.
a	a	bk

" S D : F (b ) , b (a, b):

! E I D :

7 7 7 I = S

%L'# C$" =-

∞ 2π(k+1)

2πk

k=0

sin xdx,

, : D : ,

2π(k+1)

2π(k+1)

sin xdx = − cos x 2πk

= −1 + 1 = 0.

2πk

1 cos x

∞

x

+

7

x

'

0

sin tdt D : x

0

sin tdt =

−

7

→

∞

:

, f 7 = 4 <

#"L " = 6 f C[0, +∞)

# )

∞

f (x)dx

0

0 4 ) +$ ? ( ( (

	∞

f (k) = f (0) + f (1) + . . .

k=0

#

! 7 f 7 7

k+1

f (k + 1) ≤ f (x)dx ≤ f (k).

k

77 D k: , 7

n	n+1	n+1		∞
k=0 f (k + 1) =	k=1 f (k) ≤		f (x)dx ≤ k=0 f (k).
		0

L : , : , , <

D 7 : ! E

7 $ 7 = 6: , : : ,

∞

1

k=1 kα

D α > 1 D α ≤ 1 7 : ,

(1 + x)−α α > 0 7 [0, + )
∞		= ∞ dy =					∞
	dx					1	, α > 1,
						1−α
0		1			+∞, α ≤ 1.
	(1 + x)α		yα
# 7 7			: 7 <

D , D C f : (a, b) → R: , 7

I = a	b
	f (x)dx

7 , D a b C , c (a, b) , : , 7

I1 =	c	I2 =	b
	f (x)dx,		f (x)dx

a

c

0 4 ) +$ ? ( ( (

7 , D a b

LC "5"("'$" =- / 7 : , <

I D : D I1, I2: , 7 7 <

I = I1 + I2

C 7: , M , c (a, b) C : 7: a < c < c < b #

c	b	c	c	b
	f (x)dx + f (x)dx = f (x)dx + f (x)dx + f (x)dx =
a	c	a	c	c
	c		b
	=	f (x)dx +	f (x)dx.
	a		c

7 7 ,

LC "5"("'$" = 3 C (a, b) 7 , < 7 a = c0 < c1 < · · · < cn−1 < cn = b , ,

(ck, ck+1) D: ,

ck+1

Ik = f (x)dx, k = 0, 1, . . . , n − 1

ck

7 7 (ck, ck+1) ' I 7 D 7 : D <

Ik: , 7 7

n−1

I = Ik.

k=0

! M 7 7 7 7

C$" = 3 7 7

1	x	0	x	1	x .
		=		+
	dx		dx		dx
−1		−1		0

L, : D : D : <

$7 :

−ε1		x	= ln |x|	ε1	1	x		1
				1 ,			= ln |x| ε2.
	1	dx	−		ε2	dx
			−
−

0 5 $ (

L		−ε1			1
			x +				x	= ln ε2 .
			dx				dx		ε1
		−1			ε2
C lim		ln ε1 : ε1 ε2 7
ε1,ε2	→0	ε2
7
L <
	, c				b
			(a, b) a/ 7 : , M
LC "5"("'$" = 2 C						f (x)dx Q

D " %: < ,

ε→0
	c−ε	b			b
lim	f (x)dx +	f (x)dx		= p.v.	f (x)dx.
	a	c+ε			a

$ 7 = 3 D 7 ,

% E :

1	x =	ε→0	−ε	x	1	x		= 0
					+
p.v.	dx	lim		dx		dx			.
			−1
−1					ε

/ ) ' & ' ? A

L 7 E D 7 , <

D C <

,

b

I = f (x)dx.

a

7 [a, b] n D , 7 , a = x0 < x2 < x4 < · · · < x2n = b L , 7 , x2k−1 , : <

[x2k−2, x2k]: k = 1, . . . , n: 7

0 5 $ (

I

I =	b − a	(f (x	) +	· · ·	+ f (x	2n−1	)) + R
	n	1

7 , , I

#"L " .- f C2[a, b]# ) '

η [a, b]

R = (b − a)3 f (η).

24n2

, 7

h

f (x)dx,

−h

, : , f C2[−h, h] 5 M 7

0		h
I1 =	f (x)(x + h)2dx,	I2 = 0	f (x)(x − h)2dx
−h

, 7 5 I1 7 7

I1 = (x+h)2f (x) 0 h−2				0					0
					f (x)(x+h)dx = f (0)h2−2f (0)h+2					f (x)dx.
	−	−		h			−		h
5
	I2: , : , 7
						h
	I2 = f (0)h2 − 2f (0)h + 2					0	f (x)dx.

I1 I2: , 7

h

f (x)dx = 2f (0)h = I1 + I2 .

2

−h

0 5 $ (

L		: 7 7 , : 7 7
				I	1	2		0										h	f (x)(x − h)2dx =
								2 = 2				f (x)(x + h)2dx + 2 0
						+ I		1								1
											−h
	f		(ξ )				0						f	(ξ )	h					h3		f		(ξ ) + f		(ξ )
=					1			(x + h)2dx +						2	0	(x − h)2dx =					·			1		2	.
			2											2						3				2
							−h

5 : f [−h, h]: 7 ! < E , 7 7

m = min f (ξ)	≤	f (ξ)	≤	max f (ξ) = M	ξ [	−	h, h].
[ h,h]				[ h,h]
−				−

L 7 / < % E 7 , 7 , <

, η [−h, h] : ,

f (η) =

f (ξ1) + f (ξ2)

.

2

L , 7 7

h

f (x)dx = f (0) · 2h + 24f (η) · (2h)3.

(9.1)

1

−h

! 7 I C 7

b

x2

x2n

a

f (x)dx = x

f (x)dx + · · · +x

f (x)dx,

0

2n−2

7 < 7 D 7 7 7 N.-O8

I =

b − a

(f (x

) +

· · ·

+ f (x

2n−1

)) + R +

· · ·

+ R ,

n

1

1

n

b−a = 2h 5 77 R + R +

· · ·

+ R = R 7 7

n

1

2

n

R =

(b − a)3

·

f (η1) + · · · + f (ηn)

=

(b − a)3

f (η),

24n2

n

24n2

, 7 , η [a, b]: E :

7 7 / < % E