заочникам / мат анализ / интегр исчисление

A 1

dx = m 1 (x a)m 1 + C:

x1.

•¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

11

•à¨¬¥à 21.

Z

1

Z

( 2x)

7

x2 dx = x

7

x2

x

dx = x

7

x2+

p

p

2

p7

x2

p

2

2

+ Z

p7x

x2

dx = x

Z

p77 xx2

dx + Z

p7

7

x2 dx =

7 x2

p

x

= x

p

7 x2

Z

p

7 x2

dx + 7arcsin p7

:

R

Ž¡®§- 稬 ç¥à¥§

=

. •¥à¥¯¨è¥¬ ¯®á«¥¤-¥¥ à ¢¥-á⢮ x

7 x

I

p

x2 dx

¢ ¢¨¤¥:

I = xp

7 x2

+ 7arcsin

p

I

¨«¨ 2I = xp

7 x2

+ 7arcsin

p

;

7

7

I = 2

x 7 x2 + 7arcsin p7

; ®âáî¤ ¨¬¥¥¬:

1

x

Z

p

1

7

x

7 x2 dx =

+ C:

2 x

7 x2 + 2 arcsin p7

p

p

1.4. ˆ-⥣à¨à®¢ -¨¥ à 樮- «ì-ëå äã-ªæ¨©

‘- ç « ®áâ -®¢¨¬áï - ¨-⥣à¨à®¢ -¨¨ â ª - §ë¢ ¥¬ëå ¯à®áâëå ¤à®- ¡¥©. •â® ¤à®¡¨ á«¥¤ãîé¨å ç¥âëà¥å ⨯®¢:

		A			A
I.			; II.				;
	x a				(x a)m
III.		Ax + B		; IV.			Ax + B	; (m = 2; 3; : : :):
		x2 + px + q				(x2 + px + q)m

ˆ-⥣à¨à®¢ -¨¥ ¤à®¡¥© ¢¨¤ I ¨ II ¨§¢¥áâ-®:

ZA

I. x a dx = A lnjx aj + C;

II.

Z A

(x a)m

•à¨ ¨-⥣à¨à®¢ -¨¨ ¤à®¡¥© III ¨ IV ¢ ¢ëà ¦¥-¨¨, áâ®ï饬 ¢ §- ¬¥- ⥫¥, ¢ë¤¥«ï¥âáï ¯®«-ë© ª¢ ¤à â

x2 + px + q =	x + 2				2			+ q =	x + 2			+ q	2			= x +	2		+ k:
		p	2			p	2			p	2			p	2		p	2

12

x1.

•¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

’®£¤ ¨-â¥£à « ®â ¤à®¡¨ III § ¯¨è¥âáï ¢ ¢¨¤¥:

Ax + B

Ax + B

A x + p

p

+ B

Z

dx =

Z

dx =

x +2p 2 2+ k

dx =

x2 + px + q

x + p 2

+ k

2

Z

2

p

B

A

p

B

A

p

A x + 2

2

At

2

=

dx +

dx =

dt +

dt;

t2 + k

Z

x +

p

2

+ k

Z

x +

p

2

+ k

Z

t2 + k

Z

2

2

£¤¥ t = x + 2p .

•¥à¢ë© ¨-â¥£à « «¥£ª® ¢ëç¨á«ï¥âáï ¯®¤áâ -®¢ª®© u = t2 + k,

¨¬¥--®:

t dt

1

dt2

1

d(t2 + k)

1

Z

=

Z

=

Z

=

lnjt2 + kj + C:

t2 + k

2

t2 + k

2

t2 + k

2

‚â®à®© ¨-â¥£à «

dt

ï¥âáï â ¡«¨ç-ë¬.

2

t +k

- «®£¨ç-® III ¯®«ã稬:

•à¨ ¨-⥣

à¨à®¢ -¨¨ ¤à®¡¨ IV

R

Ax + B

Ax + B dx

A x + 2p 2p

+ B

Z

dx = Z

= Z

m

dx =

(x2 + px + q)m

x +

p

2

m

p

2

2

+ k

x +

2

+ k

t dt

p

dt

= A Z

+ B A

Z

:

(t2 + k)m

2

(t2 + k)m

•¥à¢ë© ¨-â¥£à « «¥£ª® ¢ëç¨á«ï¥âáï ¯®¤áâ -®¢ª®© u = t2 + k,

¨¬¥--®:

t dt

1

dt2

1

d(t2 + k)

1

du

Z

=

Z

=

Z

=

Z

=

(t2 + k)m

2

(t2 + k)m

2

(t2 + k)m

2

um

1

1

1

1

1

=

+ C =

+ C:

2

m 1

um 1

2(m 1)

(t2 + k)m 1

‚â®à®© ¨-â¥£à « ¢ëç¨á«ï¥âáï ¬¥â®¤®¬ ¯®-¨¦¥-¨ï. •ãáâì Im

=

R

dt

,

2

m

(t +k)

⮣¤ Im 1 =

dt

.

(t2+k)m 1

•

áᬮâਬ ¨-â¥£à « I

m

:

R

I

=

dt

=

1

k

dt =

1

k + t2 t2

dt =

m

Z

(t2 + k)m

k Z

(t2 + k)m

k Z

(t2 + k)m

1

Z

k + t2

dt Z

t2

dt =

=

k

(t2 + k)m

(t2 + k)m

1

Z

dt

t dt

=

1

Im 1

1

d(t2 + k)

:

=

Z t

Z t

k

(t2 + k)m 1

(t2 + k)m

k

2

(t2 + k)m

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

13

•®á«¥¤-¨© ¨-â¥£à « ¢ëç¨á«ï¥âáï ¬¥â®¤®¬ ¨-⥣à¨à®¢ -¨ï ¯® ç áâï¬:

d t2

k

1

Z

t

( + )

=

Z

t d(t2 + k)1 m =

(t2 + k)m

m 1

1 1

(t2 + k)m 1 Im 1

:

= m 1 1 (t2 + k)m 1

(t2 + k)m 1

= m

t

Z

dt

t

Žâáî¤ ¨¬¥¥¬, çâ®

1

Im 1

1

t

Im

1

:

Im =

+

2(m

1)

k

2(m

1)

(t2

+ k)m 1

•®á«¥¤-ïï ä®à¬ã«

᢮¤¨â ¢ëç¨á«¥-¨¥ Im ª ¢ëç¨á«¥-¨î Im 1. ‡- ï ¨-â¥-

£à « (á¬. â ¡«¨æã ¨-â¥£à «®¢) I

=

R

dt

, - ©¤¥¬ I

=

dt

¨ â ª ¤ «¥¥

2

2

2

¤® ¨-â¥£à « Im.

1

t +k

2

R (t +k)

•à¨¬¥à 22.

Z

2x + 5

dx =

Z

2x + 5

dx =

Z

2(x + 2 2) + 5

dx =

x2 + 4x 3

(x + 2)2 7

(x + 2)2 7

2 (x + 2)

d(x + 2)

= Z

d(x + 2) + Z

=

(x

+ 2)2 7

(x + 2)2 7

= 2

Z

t dt

+

dt

=

Z

d(t2 7)

+

Z

dt

=

t2 7

t2 7

t2 7

Z

t2 7

p

p

= ln

t2

7

+

1

ln

7 t

+C = ln

x2+4x

3

+

1

ln

7 2 x

+C:

j

p7 + t

j

2p7

p7 + 2 + x

j

2p7

j

•à¨¬¥à 23.

Z

2x + 5

dx =

Z

2x + 5

dx = 2Z

x + 2

d(x + 2)+

(x2 + 4x 3)2

((x + 2)2 7)2

((x + 2)2 7)2

+ Z

d(x + 2)

= 2Z

t dt

+ Z

dt

=

((x + 2)2 7)2

(t2 7)2

(t2 7)2

=

Z

d(t2 7)

+

Z

dt

=

1

+

Z

dt

;

(t2 7)2

(t2 7)2

t2 7

(t2 7)2

14	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

¢ëç¨á«¨¬ ®â¤¥«ì-® ¯®á«¥¤-¨© ¨-â¥£à «:

dt

=

1

( 7)dt

=

1

t2

7 t2

dt =

Z

(t2 7)2

7

Z

(t2 7)2

7 Z

(t2 7)2

=

1

t2 7

dt +

1

t2

dt =

1

dt

+

7

Z

(t2 7)2

7 Z

(t2 7)2

7

Z

(t2 7)

+

1

t

d(t2 7)

=

1

dt

1

t d

1

=

14

Z

7 Z

t2 7

14

Z

t2 7

(t2 7)2

= 7 Z t2

7

14 t t2 7 Z

t2 7 =

1

dt

1

1

dt

p

= 14 Z

t2

7

14 t2

t

7 = 14 2p7 ln p

7

t

14 t2

7 + C:

1

dt

1

1

1

7 + t

1

t

Žª®-ç ⥫ì-® ¯®«ã稬:

ln

p

+ t

Z

2x + 5

1

1

7

1 t

dx =

+

28p

p

+ C =

(x2 + 4x

3)2

t2

7

t

14 t2

7

7

7

p

+ 2 + x

1

1

7

1

x + 2

C:

= x2 + 4x

3

2

x

14 x2

+ 4x

3 +

•à¥¦¤¥ 祬 ¯¥à¥©â¨ ª ®¡é¥¬ã á«ãç î ¨-⥣à¨à®¢

-¨ï ¯à ¢¨«ì-ëå ¤à®-

¡¥©, áä®à¬ã«¨à㥬 ®¤-ã ¨§ ⥮६

«£¥¡àë, ¨¬¥îéãî äã-¤ ¬¥-â «ì-®¥

§- ç¥-¨¥ ¢ ⥮ਨ ¨-⥣à¨à®¢ -¨ï à 樮- «ì-ëå ¤à®¡¥©: ª ¦¤ ï ¯à ¢¨«ì-

- ï ¤à®¡ì PQ((xx)) ¬®¦¥â ¡ëâì ¯à¥¤áâ ¢«¥- ¢ ¢¨¤¥ áã¬¬ë ª®-¥ç-®£® ç¨á« ¯à®áâëå ¤à®¡¥©.

• §«®¦¥-¨¥ ¯à ¢¨«ì-®© ¤à®¡¨ - ¯à®áâë¥ â¥á-® á¢ï§ -® á à §«®¦¥-¨¥¬ ¬-®£®ç«¥- Q(x) - ¬-®¦¨â¥«¨. Š ª ¨§¢¥áâ-®, ª ¦¤ë© æ¥«ë© ¬-®£®ç«¥- á ¢¥é¥á⢥--묨 ª®íää¨æ¨¥-â ¬¨ à §« £ ¥âáï - ¢¥é¥á⢥--ë¥ ¬-®¦¨â¥«¨

x

a

x2

px

q

). ‚ á奬 â¨ç¥áª®¬k

¥-¨¥ ¬-®£®ç«¥-

⨯ (

) ¨ (

+

+

¢¨¤2¥ à §«®¦m

. •®ª ¦¥¬, ª ª,

Q(x) ¬®¦-® § ¯¨á âì ¢ ¢¨¤¥: Q(x) = (x a) : : : (x +px+q)

á ãç¥â®¬ à §«®¦¥-¨ï Q(x), ¯à ¢¨«ì- ï ¤à®¡ì à ᪫ ¤ë¢ ¥âáï - ¯à®áâë¥: 1. …᫨ ¬-®¦¨â¥«ì (x a) ¢å®¤¨â ¢ à §«®¦¥-¨¥ Q(x) ⮫쪮 ¢ ¯¥à¢®© á⥯¥-¨, ¬ë ¯®áâ ¢¨¬ ¥¬ã ¢ ᮮ⢥âá⢨¥ ¥¤¨-á⢥--ãî ¯à®áâãî ¤à®¡ì:

(x a) ! x A a:

2. …᫨ ¢ à §«®¦¥-¨¥ Q(x) ¢å®¤¨â ¬-®¦¨â¥«ì (x a)k , â® ¥áâì ¯®ª § ⥫ì

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...						15
á⥯¥-¨ k > 1, â® ¥¬ã ᮮ⢥âáâ¢ã¥â á㬬				¨§ k ¯à®áâëå ¤à®¡¥©
(x a)k !	A1		A2		Ak
		+		+ : : : +		:
	x a		(x a)2		(x a)k

3. …᫨ ¢ à §«®¦¥-¨¥ Q(x) ¢å®¤¨â ¬-®¦¨â¥«ì x2 +px +q ⮫쪮 ¢ ¯¥à¢®© á⥯¥-¨, â® ¢ ᮮ⢥âá⢨¥ ¥¬ã áâ ¢¨âáï ¥¤¨-á⢥-- ï ¯à®áâ ï ¤à®¡ì:

x2 + px + q ! M x + N : x2 + px + q

4. …᫨ ¢ à §«®¦¥-¨¥ Q(x) ¢å®¤¨â ¬-®¦¨â¥«ì (x2 + px + q)k, ¯®ª § â¥«ì ª®â®à®£® k > 1, â® ¥¬ã ᮮ⢥âáâ¢ã¥â á㬬 ¨§ k ¯à®áâëå ¤à®¡¥©

(x2	+ px + q)k !	M1x + N1	+	M2x + N2	+ : : : +	Mk x + Nk	:
		x2 + px + q		(x2 + px + q)2		(x2 + px + q)k

„«ï - 宦¤¥-¨ï -¥¨§¢¥áâ-ëå ª®íää¨æ¨¥-⮢ A, M , N ¨á¯®«ì§ãîâ ¬¥â®¤ -¥®¯à¥¤¥«¥--ëå ª®íää¨æ¨¥-⮢. ‡- ï ä®à¬ã à §«®¦¥-¨ï PQ((xx)) - ¯à®áâë¥ ¤à®¡¨, ¯¨èãâ ¥£® á ¡ãª¢¥--묨 ª®íää¨æ¨¥-â ¬¨ á¯à ¢ . •à¨¢®¤ï⠤஡¨ ª ®¤-®¬ã §- ¬¥- ⥫î Q(x) ¨ ¯à¨à ¢-¨¢ îâ ¬-®£®ç«¥-ë, áâ®ï騥 ¢ ç¨- ᫨⥫ïå á¯à ¢ ¨ á«¥¢ . ‡ ⥬, ¯à¨à ¢-¨¢ ï ª®íää¨æ¨¥-âë, áâ®ï騥 ¯à¨ ®¤¨- ª®¢ëå á⥯¥-ïå, - 室ïâ -¥¨§¢¥áâ-ë¥ ¡ãª¢¥--ë¥ ª®íää¨æ¨¥-âë.

				2x2+2x+13
•à¨¬¥à 24. • §«®¦¨âì ¤à®¡ì							-	¯à®áâë¥.
				(x 2)(x2+1)2
•à¥¤áâ ¢¨¬ ¤à®¡ì ¢ ¢¨¤¥:
	2x2 + 2x + 13			A		Bx + C Dx + E
		=			+				+		:
	(x 2)(x2 + 1)2		x 2			x2 + 1				(x2 + 1)2

Š®íää¨æ¨¥-âë A, B, C, D, E ®¯à¥¤¥«¨¬, ¨áå®¤ï ¨§ ⮦¤¥á⢠:

2x2 + 2x + 13 = A (x2 + 1)2 + (Bx + C)(x2 + 1)(x 2) + (Dx + E)(x 2);

®âáî¤ ¯®«ã稬:

2x2 + 2x + 13 = (A + B) x4 + (C 2B) x3 + (2A + B 2C + D) x2+ + (C 2B + E 2D) x + A 2C 2E:

•à¨à ¢-¨¢ ï ª®íää¨æ¨¥-âë ¯à¨ ®¤¨- ª®¢ëå á⥯¥-ïå x á«¥¢ ¨ á¯à ¢ , ¯à¨¤¥¬ ª á¨á⥬¥ ¨§ ¯ï⨠ãà ¢-¥-¨©

8

>A + B = 0;

>

> 2B + C = 0;

>

<

2A + B 2C + D = 2;

> 2B + C 2D + E = 2;

>

>

>

: A 2C 2E = 13;

®âªã¤

A = 1; B = 1; C = 2; D = 3; E = 4:

16				x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...
Žª®-ç ⥫ì-®:
	2x2 + 2x + 13	1			x + 2			3x + 4
		=							:
	(x 2)(x2 + 1)2		x 2		x2 + 1			(x2 + 1)2

„«ï ⮣® çâ®¡ë ¯à®¨-⥣à¨à®¢ âì ¯à ¢¨«ì-ãî ¤à®¡ì, ¥¥ à ᪫ ¤ë¢ îâ

- á㬬㠯à®áâëå ¤à®¡¥©,

§ ⥬ ¨-⥣à¨àãîâ ª ¦¤ãî ¯à®áâãî ¤à®¡ì

®â¤¥«ì-® ¨ ¨å १ã«ìâ âë ᪫ ¤ë¢ îâ.

•à¨¬¥à 25.

2x2 + 2x + 13

dx

x + 2

3x + 4

Z

dx = Z

Z

dx Z

dx =

(x 2)(x2 + 1)2

x 2

x2 + 1

(x2 + 1)2

=

1

3 4x

+

1

ln

(x 2)2

4arctgx + C:

2 x2 + 1 2

x2 + 1

‡¤¥áì ¬ë ¢®á¯®«ì§®¢ «¨áì à §«®¦¥-¨¥¬ ¨§ ¯à¨¬¥à

24.

1.5. ˆ-⥣à¨à®¢ -¨¥ ¨àà 樮- «ì-ëå ¢ëà ¦¥-¨©

‚ëè¥ ¬ë - ã稫¨áì ¨-⥣à¨à®¢ âì à 樮- «ì-ë¥ äã-ªæ¨¨. ‡¤¥áì à á- ᬠâਢ ¥âáï ¬¥â®¤ à 樮- «¨§ 樨 ¤«ï ¨-⥣à¨à®¢ -¨ï ¨àà 樮- «ì-ëå ¢ëà ¦¥-¨©. € ¨¬¥--®, ¨é¥âáï ¯®¤áâ -®¢ª t = t(x), ª®â®à ï ¯à¨¢¥« ¡ë ¨àà 樮- «ì-®¥ ¢ëà ¦¥-¨¥ ª à 樮- «ì-®¬ã ¢¨¤ã. ‡¤¥áì ¨ ¤ «ìè¥ ¡ã¤¥¬ ¯®« £ âì, çâ® R(x; y; z; : : :) | à 樮- «ì- ï äã-ªæ¨ï ®â ᢮¨å à£ã¬¥-⮢.

I. ˆ-⥣à¨à®¢ -¨¥ ¢ëà ¦¥-¨© ¢¨¤ :

					Z R x; s			cx + p	! dx;
							m ax + b
£¤¥ m | - âãà «ì-®¥ ç¨á«®, a, b, c, p | ¯®áâ®ï--ë¥. •®«®¦¨¬:
s		cx + p				cx + p				a ctm
t = t(x) = m		ax + b	;		tm =	ax + b	; x = '(t) =			p tm b	; dx = '0(t)dt:

ˆ-â¥£à « ¯¥à¥¯¨è¥âáï ¢ ¢¨¤¥:

Z

R('(t); t)'0(t)dt:

‚ëç¨á«¨¢ ¯®«ãç¥--ë© ¨-â¥£à «, ¢¥à-¥¬áï ª áâ ன ¯¥à¥¬¥--®© t = t(x).

•à¨¬¥à 26.

Z x + 1 rx 1 dx:
1	3 x + 1

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

17

•®« £ ¥¬ t = q

x 1

, x = t3

1, dx =

(t3 1)2 dt, ⮣¤

3 x+1

t3

+1

6t2

Z x + 1 r

x 1

dx = Z

t3 1 = Z

t 1 + t2

+ t + 1 dt =

1

3 x + 1

3dt

1

t + 2

1 t2 + t + 1

p

2t 1

2

(t 1)2

p3

=

ln

+ 3 arctg

ˆq

x+1

¦¥-¨© ¢¨¤ :

£¤¥ t =

3

.

x 1

II.

-⥣à¨à®¢ -¨¥ ¢ëà

Ax

B

Z

+

dx:

p

ax2 + bx + c

1) •ãáâì a > 0, ⮣¤

¨-â¥£à « (3) ¯¥à¥¯¨è¥âáï ¢ ¢¨¤¥

Z

Ax + B

1

Z

Ax + B

dx =

p

dx:

p

x2 + b

x + c

a

p

x2 + px + q

a

q

a

a

+ C;

(3)

’ ª ¦¥, ª ª ¨ ¢ á«ãç ¥ ¨-⥣à¨à®¢ -¨ï à 樮- «ì-®© äã-ªæ¨¨, ¢ë¤¥- «¨¬ ¯®«-ë© ª¢ ¤à â ¢ ª¢ ¤à â-®¬ âà¥åç«¥-¥, áâ®ï饬 ¢ §- ¬¥- ⥫¥ (á¬. ¯ã-ªâ 1.4):

1

B

1

A x + p

Z

Ax +

dx =

22

dx+

p

p

a

p 2

a

p

q x + 2

+ k

p

Z q x +

2

+ k

1

1

B A

2

1

At

+

p

Z

dx =

p

Z

p

dt +

p

Z

a

q

x + p

2

+ k

a

t2 + k

a

£¤¥ t = x + p .

2

2

•¥à¢ë© ¨-â¥£à « ¢ëç¨á«ï¥âáï ¯®¤áâ -®¢ª®© u = t2 + k

t dt

1

dt2

1

d t2 + k) 1

du

Z

=

Z

=

Z

(

=

Z

=

p

2

p

2

p

2

p

t2 + k

t2 + k

t2 + k

u

1

Z

u 1=2 du = u1=2 + C = (t2

=

2

B A 2p
pt2 + k	dt;

+ k)1=2 + C:

‚â®à®© ¨-â¥£à « R ptdt2+k ï¥âáï â ¡«¨ç-ë¬.

18	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

2) •ãáâì a < 0, ⮣¤ ¨-â¥£à « (3) ¯¥à¥¯¨è¥âáï ¢ ¢¨¤¥

				Ax			B
Z						+
	p
				q		x2	b	x	c
			a
						a			a

1

Ax

B dx

dx =

Z

+

=

p

p

a

x2 px

q

1

Z

Ax + B

=

dx:

p

p

a

(x2

+ px + q)

„ «ìè¥, ¢ë¤¥«ïï ¯®«-ë© ª¢ ¤à â ¢ ª¢ ¤à â-®¬ âà¥åç«¥-¥ ¨ ¯à¨¬¥-ïï § -

¬¥-ã t = x + p2, â ª ¦¥ ª ª ¨ ¢ ¯à¥¤ë¤ã饬 á«ãç ¥ ¢ëç¨á«ï¥¬ ¯®«ãç¥--ë© ¨-â¥£à «.

•à¨¬¥à 27.

Z

5x 1

dx =

Z

5x 1

dx =

px2 + 2x + 2

(x + 1)2

+ 1

Z

5t

6p

t dt

= Z

p

dt = 5Z

p

6

t2 + 1

t2 + 1

5(x + 1 1) 1							dx =
p(				dt
			x + 1)2 + 1
Z	p				=
				t2 + 1

5

Z

d(t2 + 1)

=

p

6 lnjt + t2

+ 1j + C =

2

t2 + 1

5

p

=

2(t2 + 1)1=2 6lnjt + pt2 + 1j + C = 5 (x2 + 2x + 2)1=2

2

6lnjx + 1 + p

x2 + 2x + 3

j + C:

•à¨¬¥à 28.

Z

p

5x + 11

dx = Z

5x + 11

dx =

6x x2 5

(x2 6x + 5)

dx =

=

5

x

+p

dx =

5(x

Z

11

Z

3 + 3)+ 11

((x 3)2 4)

4 (x 3)2

p5t + 26

t

p

dt

= Z

p

dt = 5Z

p

dt + 26Z

p

=

4 t2

4 t2

4 t2

5

dt2

t

5

d(4 t2)

=

Z

p

+ 26arcsin

+ C =

Z

p

+

2

2

2

4

t2

4 t2

t

+ C = 5p

+ 26arcsin

t

+ C = x

+ 26arcsin

4 t2

2

2

3

p

2

=

5

6x

x2

5 + 26arcsin

+ C:

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

19

III. ˆ-⥣à¨à®¢ -¨¥ ¢ëà ¦¥-¨© ¢¨¤ :

Z (Ax + B) p

dx:

ax2 + bx + c

•а¨ ¨-в¥£а¨а®¢ -¨¨ нв¨е дг-ªж¨© ¨б¯®«м§говбп ¨-в¥£а «л

Z

1

+ x2 + a2 lnjx + a2 + x2j) + C;

a2

+ x2 dx =

(x

a2

2

p

1 p

p x

a2

+ C;

Z

x2 dx = 2

x

a2 x2 + a2 arcsin a

p

p

ª®в®ал¥ ¢лз¨б«повбп ¬¥в®¤®¬ ¨-в¥£а¨а®¢ -¨п ¯® з бвп¬ (б¬. ¯г-ªв 1.3).

p

„«ï ¢ëç¨á«¥-¨ï ¨-â¥£à «®¢

(Ax + B)

2

+ bx + c dx ¢ ª¢ ¤à â-®¬

ax

âà¥åç«¥-¥ ¢ë¤¥«ï¥âáï ¯®«-ë© ª¢R

¤à â,

§ ⥬,

- «®£¨ç-® á«ãç î II, ¢á¥

᢮¤¨âáï ª ¢ëç¨á«¥-¨î ¨-â¥£à «®¢ ¢¨¤

p

2

dt ¨

p

2

dt. •¥à-

¬:

t k t

k t

R

¢ë© ¨§ ¯®«ãç¥--ëå ¨-â¥£à «®¢ ¢ëç¨á«ï¥ R

1

1

Z t k t2 dt =

Z

k t2 dt2

=

Z

k t2 d(k t2) =

2

2

p

p

1

2 p

1

=

(k t2)3=2

=

(k t2)3=2 + C:

2

3

3

‡- ç¥-¨¥ ¢â®à®£® ¨-â¥£à « á¬. ¢ëè¥.

•à¨¬¥à 29.

Z (2x 1)

dx = Z (2x 1) (x2 3x)dx =

3x x2

p

p

= Z (2x 1)s

x 2

4 dx =

3

2

9

= Z 2 x 2

+ 2 1 s

4 x 2

dx = Z (2t + 2)r4 t2 dt =

3

3

9

3

2

9

= 2Z tr

4 t2

dt + 2Z r

4 t2

dt = Z r

4 t2

dt2+

9

9

9

! = 3 (3x x2)3=2+

+ 2 2

tr

4

t2 + 4

arcsin 3

1

9

9

2t

2

3

+ x

2

p3x x2 + 4 arcsin

+ C:

3

9

2x

3

20	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...
IV. ˆ-⥣à¨à®¢ -¨¥ ¤¨ää¥à¥-æ¨ «ì-ëå ¡¨-®¬®¢ ¢¨¤ :
Z	xm(a + bxn)p dx;

£¤¥ p = ; , | æ¥«ë¥ ç¨á« ; m, n | ¯à®¨§¢®«ì-ë¥ ç¨á« .

„ --ë© ¨-â¥£à « ¨-⥣à¨àã¥âáï «¨èì ¢ á«¥¤ãîé¨å âà¥å á«ãç ïå:

1)¥á«¨ p 楫®¥ ç¨á«®, â® ¨á¯®«ì§ã¥âáï ¯®¤áâ -®¢ª

x = tN ; £¤¥ N | ®¡é¨© §- ¬¥- â¥«ì ¤à®¡¥© m ¨ n;

2)¥á«¨ mn+1 楫®¥ ç¨á«®, â® ¨á¯®«ì§ã¥âáï ¯®¤áâ -®¢ª

a+ bxn = t ;

3)¥á«¨ mn+1 + p 楫®¥ ç¨á«®, â® ¨á¯®«ì§ã¥âáï ¯®¤áâ -®¢ª

a + bxn = xnt :

•à¨¬¥à 30.

Z

p

x3

dx;

x2 1

£¤¥ m = 3, n = 2. —¨á«®

m+1

= 3+1

= 2 | 楫®¥, ¯®í⮬㠨ᯮ«ì§ã¥¬

n

2

¯®¤áâ -®¢ªã (1): x2 1 = t2, x = p

t

t2 + 1, dx =

p

dt, ®âáî¤

1+

t2

x3

t2

)

3=2

t

Z

p

dx =

Z

(1 +

dt = Z (1 + t2)dt =

t

(t2 + 1)1=2

x2 1

= t +

t3

+ C = (x2

1)1=2 +

(x2

1)3=2

+ C:

3

3

•à¨¬¥à 31.

Z

p4

dx

= Z

x0(1 + x4) 1=4 dx;

1 + x4

§¤¥áì m = 0, n = 4, p = 41. •à®¢¥à¨¬ ãá«®¢¨¥:

m+1

+ p | 楫®¥ ç¨á«® ¨«¨

n

-®«ì. mn+1+p = 0+14 41 = 0, ¯®í⮬ã4

¨á¯®«ì§ã¥¬ ¯®¤áâ -®¢ªã (2); §¤¥áì = 4:

p

4

4

4

4 1

1+x4

4

1=4

3

4

5=4

1+ x

=p4

t

,

4

= q

+ 1 =4

x

1,=4

= (

t

1)

,

dx

=

t

(

t

1)

dt

,

x

4

x

t

x

= tx = t (t

1)

â ª çâ®

1 + x

,

dx

t2 dt

1

1

1

1

dt

Z

p4

= Z

=

Z

dt

Z

=

t4 1

4

t + 1

t 1

2

t2 + 1

1 + x4

= 4 ln t

1

2 arctgt + C;

1

t + 1

1

4

4

p1+x

£¤¥ t =

.

x