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Национальный исследовательский университет «МЭИ»

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Высшая математика

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УрМатФиз / УрМатФиз с теорией

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Rx t =ch x sh t + xt + t 2 h 1 sh 1 ch x +

3x!

j j 3=2:

3 2

" # $ % f x =x

f1 = Z1 sh s s ds = 2 ch 1 sh 1 f2 = Z1 ss ds = 2=3:

&' % ( #

# j j 3=2 # %

yx = 2 h 1 sh 1 ch x +

3x+

2 h 1 sh 1 ch x +

3x!!+ x=

3 2

2 h 1 sh 1 ch x +

3x! + x:

3 2

1 $ ) $ )

i = 1 1:

x t = arctg x t + t

2 1!

($ :

Rx t = arctg x t + t2

j j 2= 2 :

2 2

yx =

arctg x + x

j j 2= 2 :

3 2

2 $ ) $ )

Kix t = t 2 h 1 sh 1 ch x + 32x! 32!i 2 i =

2 1

($ :

Rx t =ch x sh t + xt + t 2 h 1 sh 1 ch x + 3x!

3 2

j j 3=2:

2 h 1 sh 1 ch x +

yx =

3x! + x # j j 3=2:

3 2

311


7.2.			II
7.1		K x t f x .
			K x t .
R x t !										,
, # $ %						$ .
& !			R x t							' ().
		K x t = x=p
1.	1			1		t2 a = 0 b = 1 f x = x:
	2							b = 2 f x = x:
		K x t = cos x + t a = 0
2.	1	K x t = x sin 2t				a = 0	b =		f x = x:
	2	K x t = sin cos t + xt					a =		b = f x = x:
3.	1 K x t = sh x ch t a = 0 b = 1 f x = x:
	2	K x t = cos x t a = 0						b = 2 f x = x:
4.	1 K x t = ch x				t	1 a = 0 b = 1 f x = sh x:
	2	K x t = cos sin t + xt					a =		b = f x = x:
5.	1	K x t = cos 2x t a = 0 b = =2 f x = sin x:
	2 K x t = ch x + t a = 1 b = 1 f x = 1:
6.	1 K x t = t= 1 + x2 a = 0 b = 1 f x = x:
	2 K x t = sh x ch t + ch x a = 1 b = 1 f x = 1:
7.	1	K x t = sin x cos 2t a = 0 b = f x = sin x:
	2 K x t = ch x t a = 1 b = 1 f x = 1:
8.	1	K x t = cos x sin t a = 0						b = =2 f x = x:
	2 K x t = ch x sh t + ch t a = 1 b = 1 f x = x:
9.	1	K x t = ln x=t				a = 1	b = 2 f x = ln x:
	2	K x t = sin x cos t + x sin t						a = b =		f x = x:
10.	1	K x t = arctg x t a = 0						b = 1 f x = x:
	2	K x t = cos x sin t + sin x t a = b = f x = x:
11.	1	K x t = ex t 1 a = 0 b = 1							f x = ex:
	2 K x t = ch x sh t + sh x t a = 1 b = 1 f x = x:
12.	1	K x t = arcsin x				t a = 0		b = 1 f x = x:
	2 K x t = sh x ch t + x sh t a = 1 b = 1 f x = x:

312

13.

1 K x t = x= 1 + t2 a = 0 b = 1 f x = x:

K x t = cos x

t + t2

a =

b =

f x = x:

14.

K x t = x

1 sh t

a = 0

b = 1 f x = x:

K x t = sin x

t + sin t

a =

b =

f x = x:

15.

K x t = x

1 ch t

a = 0

= 1 f x = x:

K x t = x sin t + sin x

a =

b =

f x = x:

16.

K x t = 1= x ln t

a = 2

= 3

f x = x ln x:

K x t = sin x

t + x

a =

b =

f x = x:

17.

K x t = x ln t a = 1 b = 2

f x = x:

K x t = x sin t + t

a =

b =

f x = x:

18.

K x t = x arctg t

a = 1

b = 1 f x = x:

2 K x t = sh x t + sh t

a = 1 b = 1 f x = x:

19.

K x t = x arcsin t

a =

b = 1 f x = x:

2 K x t = x sh t + sh x a =

1 b = 1 f x = x:

20.

K x t = 1= 1 + x2

a = 0

b = 1 f x = x:

2 K x t = sh x t + x a = 1 b = 1 f x = x:

21.

K x t = 1

x et

a = 0

b = 1 f x = x:

2 K x t = x sh t + t a =

1 b = 1 f x = x:

22.

K x t = sh x

t 1

a = 0 b = 1

f x = ch x:

K x t = cos x

t2 + x2 sin t

a =

b = f x = x:

23.

K x t = ch x

1 t

a = 0

b = 1 f x = ex:

2 K x t = x2t2 + sh x t a = 1 b = 1 f x = x:

24.

K x t = t=p

a = 0

b = 1 f x = x:

K x t = x2t2 + x sin t

a =

b =

f x = x:

25.

K x t = 1 + x2 sin 2t

a = 0

b =

f x = x:

K x t = x2t2 + sin x

a =

b =

f x = x:

26.

K x t = x 1 t2

a = 0

b = 1 f x = ex:

2 K x t = xt + ch x t2 a =

1 b = 1 f x = x:

27.

K x t = cos x

t2 + 1

a = 0

b = =2

f x = sin x:

K x t = cos x cos t + xt

a =

b =

f x = x:

313

28.	1	K x t = sin x t + 1 a = 0 b = =2 f x = cos x:
	2 K x t = ch x t2 + sh x t a = 1 b = 1 f x = x:
29.	1	K x t = 1 + t =1 + x2 a = 1 b = 1 f x = x:
	2 K x t = x2 ch t + x sh t a = 1 b = 1 f x = x:
30.	1	K x t = arcsin x t + 1 a = 1 b = 1 f x = x:
	2 K x t = x2t2 + x sh t a = 1 b = 1 f x = x:

7.3.

7.1

K x t = K t x 2 C a b :

|. ! " -"$ $ % " " & .

I. '()$ "$ & " *$ + " * ) , + % " " % *.

II. ! % " " * ) -, ". /- .

III. 0 ) -, ". , ,(/ &* % "- " * i 6= j , 2 *3 a b :

b	b
Z	yi x yj x dx = ij Z	yi2 x dx:
a	a

IV. 5 + , % " " , *, , -

* & *$ & % )- % -, ".$. 6 -, ". + 3 2 *3. 2 * & . 7 |9 .

. % " " * " & ,*( " + * *3" &, " " 2 " 3. 6 * *3 3 & -

, $ , -.

314

| .

, !-

" #

$a b% &

y x = f x +

yn x

:3:1

yn x | !

+ ,-, fn | .++ , /

fn = Z

f x yn x dx:

:3:2

1 | !

. 1 2 ,

" 2

Z f x yn x dx = 0 n =

1 k

:3:3

y1x

::: yk

x | !

+ , , "# -

1 k. 5 6 "

yx = f x +

yn x +

Cnyn x

:3:4

n=k+1 n

n=1

C1 ::: Ck | 2

$6, 8, 9%.

7.3. : / II

Kx t = sin x cos

t = cos 1

0 x t

:3:5

cos

x sin t= cos 1

t x 1

2 : f1 x = p

sin x=2 f2

x = 1:

=- "

" !

+ ,- .

! , 2 , , 2 &

, 2 " 2,2 =- , -.

315

. 7.3.5 7.2 ,

y x =

cos 1

x Z sin t y t dt +sin x Z cos 1

t y t dt :

7:3:6

cos 1

"" # 7.3.6 :

y x =

sin 1

x Z sin t y t dt + cos 1

x sin x y x +

cos 1 B

t y t dt sin x cos 1 x y x 1 =

+ cos x Z cos 1

sin 1

x Z

sin t y t dt + cos x Z cos 1

t y t dt

7:3:7

cos 1

y x =

cos 1

x Z sin t y t dt + sin 1

x sin x y x

cos 1

			1
		sin x Z		cos 1 t y t dt	cos x cos 1
		0	x
		0		x	1
=		@	cos 1	x Z sin t y t dt	+sin xZ cos 1
=			cos 1	x Z sin t y t dt	+sin xZ cos 1

	cos 1B			0	x

x y x 1 =

t y t dt1 y x : 7:3:8

7.3.6 7.3.8

y + + 1 y = 0:

7:3:9

! 7.3.6 x = 0

7.3.7

x = 1 ( )

y 0 = 0

y 1 = 0:

7:3:10

7.3.9 , 7.3.10 ./

7.2 , 7.3.5 . 1 2

7.3.9 , 7.3.10

. 1

. . 4 5 ( !

l = 1 !

" 3 1.22

n + 1 = 2n + 1 =2 2 n =

7:3:11

0 1

5 ( " /#

1.23

yn x = p

sin 2n + 1 x=2

n =

0 1

316

					7.3.11 .
f x = f1 x = p2 sin x=2 ! "##$ !
% 7.3.2 :
						f0 = 1 fn = 0 n =									1 1:
( )				*+ , f1 x =											p			sin x=2 #
( )				*+ , f1 x =											p	2		sin x=2 #
-, 7.3.1 , . ,:
y x = p							4p			sin x=2 =							p
		sin x=2 +						2											2 2		4	sin x=2:
	2							2											2 2
	2
						2	4 + 1								2					4	+ 1
f x = f2 x = 1 ! "##$ ! %
7.3.2 :
					1
					1									2p2

fn = p2 Z sin							2n + 1x! dx =

						2														n	= 0 1:
						2							2n + 1							n	= 0 1:
					0
( )			*+ , f2 x = 1 #																		-, 7.3.1
,
			16		1				1
					X
y x = 1 +					X									4	+ 1 sin 2n + 1x=2:
y x = 1 +					n=0 2n + 1 22n + 12									4	+ 1 sin 2n + 1x=2:
																					2 = 25 2=4 1.
f x = f1 x =									p		2	sin x=2									)

7.3.3 ! :

Z sin x=2y2 x dx = 2 Z sin x=2 sin 5 x=2 dx = 0:

( ) *+ , # 7.3.4

y x = f x +

yn x + Cy2 x = p2 sin x=2 + C sin 5 x=2:

n=0

X=2

f x = f2 x = 1 ) 7.3.3

! !:

2p2

p2 Z

6=:0

1 sin 5 x=2 dx = 5

* 3 = 2 = 25 2=4	1 ,
f2 x = 1 ) .

317

f1 x

7.3.

n = 2n+1

yn x

p2 sin 2n+1 x=2

n = 0 1:

p2 sin x=2

6=n f1 x

y x

sin x=2 :

2 4

+ 1

f2 x

y x+

sin 2n

+ 1 x=2 :

n=0 2n + 1

22n + 1 2 4

+ 1

2 = 25 2=4 1 f1 x p

sin x=2

sin x=2 + C sin 5 x=2 :

y x

f2 x

= 1 .

! " #$ II 7.1 & &$$$&$ $ K x t K t x ' &$ &$ $ f1 x f2 x

( $)$# *+ $*+ $* &, -* ). .

, $ , ) )$ '$, # , #'* 0$.

, $ , ) )$* ' +, # *-'$. ( , * & &.

1.K x t

2.K x t f2 x1

3.K x t f2 x1

4.K x t f2 x

11 xt	0t	xx	t12

cos x sin 1		t = cos 1
sin 1 x		os t= cos 1
:

	os 3 x=2		f2 x :
0	x t	f1 x	= cos 5 x=2
t	x 12

sin x sin 1		t = sin 1	0	x t	f1 x	= sin 3 x
sin 1	x	in t= sin 1	t x 12
:
cos x cos 1		t = sin 1	0	x t	f1 x	os 4 x
cos 1	x	os t= sin 1	t	x 12
x:

318

	sh x sh 1 t = sh 1 0 x t
5.	K x t = sh 1 x sh t= sh 1			t x 1		f1 x = sin 3 x
	f2 x = 1:
	ch x sh 1 t = ch 1 0 x t
6.	K x t = sh 1 x ch t= ch 1			t x 1		f1 x = cos 5 x=2
	f2 x = 1:
	sh x ch 1 t = ch 1 0 x t
7.	K x t = ch 1 x sh t= ch 1			t x 1		f1 x = sin 7 x=2
	f2 x = 1:
	ch x ch 1 t = sh 1 0 x t
8.	K x t = ch 1 x ch t= sh 1			t x 1		f1 x = cos 5 x
	f2 x = x:
	x 0 x t
9.	K x t = t t x 1		f1 x = sin 5 x=2 f2 x = 1:
	x 1 t 0 x t
10.	K x t = t 1 x t x 1				f1 x = sin x f2 x = 1:
	sin x cos t	0 x t
11.	K x t = cos x sin t	t x =2 f1 x = sin 4x f2 x = 1:
	cos x sin t	0 x t
12.	K x t = sin x cos t	t x			f1 x = cos 3x=2 f2 x = 1:
	cos x sin t 2 = cos 2				0 x t
13.	K x t = sin x 2 cos t= cos 2				t x 2	f1 x = cos 3 x=4
	f2 x = 1:
	sin x sin t 2 = sin 2				0 x t
14.	K x t = sin x 2 sin t= sin 2				t x 2	f1 x = sin x=2
	f2 x = 1:
	sh x sh t 2 = sh 2 0 x t
15.	K x t = sh x 2 sh t= sh 2			t x 2		f1 x = sin x=2
	f2 x = 1:
	ch x sh t 2 = ch 2 0 x t
16.	K x t = sh x 2 ch t= ch 2			t x 2		f1 x = cos 3 x=4
	f2 x = 1:
	t 2 0 x t
17.	K x t = x 2 t x 2			f1 x = cos 3 x=4 f2 x = 1:

319

	x t 2 =2 0 x t
18.	K x t = t x 2 =2		t x 2		f1 x = sin x				f2 x = 1:
	sin x cos t		0 x t
19.	K x t = cos x sin t		t x		f1 x = sin x=2 f2 x = 1:
	cos x sin t	=2 x t
20.	K x t = sin x cos t	t x =2					f1 x = sin		x +	2 ! =2!
	f2 x = 1:
	sh x + 1 sh 1 t = sh 2 1 x t
21.	K x t = sh t + 1 sh 1 x = sh 2					t x 1
	f1 x = sin x + 1		f2 x = 1:
	ch 1 + x sh 1 t = ch 2 1 x t
22.	K x t = sh 1 x ch 1 + t = ch 2					t x 1
	f1 x = cos x + 1 =4		f2 x = 1:
	sh x + 1 ch t 1 = ch 2 1 x t
23.	K x t = ch x 1 sh t + 1 = ch 2					t x 1
	f1 x = sin x + 1 =4		f2 x = 1:
	ch 1 + x ch t 1 = sh 2 1 x t
24.	K x t = ch 1 + t ch x 1 = sh 2					t x 1
	f1 x = cos x 1		f2 x = x:
	x + 1 1 x t
25.	K x t = t + 1 t x 1			f1 x = cos x 1 =4						f2 x = 1:
	x + 1 t 1 =2 1 x t
26.	K x t = t + 1 x 2 =2			t x 1				f1 x = sin x + 1
	f2 x = 1:
	sin x cos t	=2 x t
27.	K x t = cos x sin t	t x =2					f1 x = sin x 2 !, 2!
	f2 x = 1:
	sin x cos t		x t
28.	K x t = cos x sin t		t x			f1 x = sin x + =4
	f2 x = 1:
	t x t
29.	K x t = x t x				f1 x = sin x=4				f2 x = 1:
	cos x sin t		x t
30.	K x t = sin x cos t		t x			f1 x = sin x + =2

f2 x = 1:

320

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