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Высшая математика

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. .

11.03.01, 11.03.02, 09.03.01, 09.03.02

2017

517

ББК: 22.11я7 П-691

	. .,	. .,	. .,	. .,
.	.			.
.				.
	.		.	:

/. – Барнаул: Новый формат, 2017. – 148 .

				2013 .
	210700,	230100,	230400,	210400.
		.		,
			,	,
			,
	.
	,			.
. 45,	. 4	.

27.03.2017 .				.	№
27.03.2017 .
.	: . .	, . .- . .,
	: . .	, . .- . .,		(	)
	. .	, . .- .	.,	(	C )

........................................................................................................................1			7

1.	............................................................................. 8
1.1.	................................................................................................ 8
1.2.	.	-	................ 8
1.3.	,		............... 8
1.4.		............................................................. 11
2
2.			11
		........................................	11
2.1.			(
)		..................................................................	11
2.2.			14
2.3.	............................................................................................		14
2.3.		............................................................. 16
3
3.			17
	.........		17
3.1.	.................................................................................. 17
3.2.	............................................................................... 18
3.3.	,		............. 19
3.4.		............................................................. 23
4
4.			24
	.......................		24
4.1.	........................................................................................................... 24
4.2.	....................................................................... 25
4.3.	,		............. 25
4.4.		............................................................. 29
5
5.			..... 30
5.1.	............................................................................................... 30
5.2.
5.3.	.........................................................................		31
5.3.			32
....................................................................................................................	,		32
5.4.	,		............. 33
5.5.		............................................................. 36
6
6.			37
..................................................................................................................			37
6.1.	............................................................................................... 37
6.2.
	.........................................................................		38

6.3.			39
		......................................................................................................................,	39
6.4.			............. 40
6.5.		7	............................................................. 43
7.	.	7	.................... 44
7.	.	8	.................... 44
8.		8
8.		........................................................................................ 45
8.1.			.................................................. 45
8.2.		,	........................ 46
8.3.		,	............. 47
8.4.		9	............................................................. 53
9.		9	.
9.			.
		..........................................................................................................	54
9.1.			........................................................ 54
9.2.		,	............................ 54
9.3.		,	............. 55
9.4.		10	............................................................. 59
10.		10
10.		....................................................................................... 61
10.1.		.................................................................... 61
10.2.			..................................................... 61
10.3.		,	........... 63
10.4.		11	........................................................... 68
11.		11	.............. 69
11.			.............. 69
11.1.		,	............. 69
11.2.		,	........... 70
11.3.		12	........................................................... 75
12.		12	.................................... 76
12.			.................................... 76
12.1.		................................................................. 76
12.2.		,	........................................................ 77
12.3.		,	........... 77
12.4.		13	........................................................... 81
13.		13	.................................... 82
13.			.................................... 82
13.1.		,	........................................................ 82
13.2.		,	........... 83
13.3.		14	........................................................... 87
14.		14
14.		................................................................... 88
14.1.			....................................................... 88
14.2.			....................................................... 89
14.3.		.................................................................................................. 90

14.4.		,		........... 90
14.5.		........................................................... 95
		15
15.		.	-	96
		.................................................................................................		96
15.1.		.......................................................................... 96
15.2.			.	-
		...........................................................................................................		96
15.3.		,		........... 97
15.4.		......................................................... 100
		16
16.		.	.................................. 100
16.1.		.................................................................................... 100
16.2.		.	.......................................... 101
16.3.		,		......... 102
16.4.		......................................................... 105
17.		17
17.		............................................................ 105
17.1.
		................................................................................................................		105
17.2.		..................................................................... 106
17.3.		................................................................... 107
17.4.		,		......... 108
17.5.		......................................................... 111
18.	.	18	..................................... 112
18.	.	19	..................................... 112
19.		19	(	)
19.			(	)
		..............................................................................................		113
19.1.		........................................................................................... 114
19.2.		,	........................................................ 114
19.3.		,		......... 115
19.4.		......................................................... 116
19.5.		.	........................................ 117
19.6.		......................................................... 119
20.		20
20.				119
20.1.		................................................................................................................		119
20.1.				............... 119
20.2.		......................................................... 121
20.3.		.	............................................ 121
20.4.		......................................................... 122
21.		21	.	,
21.			.	,
		.	....................................................	122
21.1.		.		............... 122

21.2.	,			.
21.3. ...........................................................................................				124
	......................................................... 127
22
22.	,			128
......................................................................................				128
22.1.			m 1	m n .
22.1.	y, y , y ,...y			m n .
..........................................................................................................		x .		128
22.2.		x .		..... 129
22.3.	......................................................... 131
23
23.				132
	..........................................................			132
23.1.	,			......... 132
23.2.	......................................................... 134
24
24.
.......................................................................................				134
24.1.	,			......... 135
24.2.	......................................................... 136
25
25.
(	).............................................................			137
25.1.	,			......... 137
25.2.	......................................................... 139
26
26.				......................... 140
26.1.	,			......... 140
26.2.	......................................................... 142
27
27.				143
...................................................................................				143
27.1.	,			............................ 143
27.2.	,			......... 143
27.3.	......................................................... 147
........................................................................................				147

	«
.	.	.
	.	.
»
«		».
		Д1Ж.
	,
		,
	,
		.
		,
.
.	,
	.
,		,
Д2Ж – [4].

1.1.

f x dx

1.1.1.

i xi , n 1, 2, 3...

max xi 0([a,

i 1

xi ,

i [xi 1, xi ] , xi xi xi 1 , x0 a , xn b ).

i xi .

dx lim f

0 i 1

1.2.

b a .

f (x)dx 0 . 1.2.3.

f x dx f x dx.

1.2.1.

1.2.2.

x dx. 1.2.5.

1.2.4.

x f2 x dx f1 x dx f2

x dx f x dx.

x dx

t dt .

1.2.6.

f x dx f

x dx

f x dx. 1.2.7. f

1.2.8.

F x

f x ,

f x dx

F x

b F b F a .

1.3.

1.3.1.

x2 2x

1				dx					1						dx						1							dx	1			d x 1

																									x 1 2							x 1 2
1			x2	2x 2					1			x2 2x						1 1			1								1 1								1
1		3	x2	2x 2					1	3		x2 2x						1 1			1		3						1 1		3						1
arctg x 1							1		arctg 0 arctg 1															1 arctg 0 arctg											0 .
arctg x 1							1		arctg 0 arctg 1													3		1 arctg 0 arctg										3	0 .
							1	3																													3	3
							1	3

											x 1 3									1 8x2
									27		x 1 3						x			1 8x2
1.3.2.																3 x										dx.
									1
				.																																		,
	x 1 3					1 8x2															:
																														2			1			5
					x									3		3											2
					x								x	3	x	3			x x 1						8x		2
													x		x				x x 1						8x			x 1 x 3 x 3 8x3 .

				3 x																3 x
				3 x																3 x

																																	1.2.3			1.2.4
		x 1 3							1 8x2									-					1.2.8:
																				2				1				5							27 2
27							x								27					2				1				5		27			27		27 2
							x
						3
												dx				x 1 x 3					x			3		8x3 dx					xdx			dx		x 3 dx
1								x							1															1			1		1
27			1					27 5					2	27						5		27				2	27				8	27
																				5		27				2	27				8	27
												x							3x	3					3x	3				3x	3
x				3	dx 8 x					3	dx				x		127												8


												2						5					2							8
	1							1				2		1				5					2							8
	1							1						1								1					1					1

12 272 1 27 1 53 35 1 32 32 1 3 38 1

364 26 145,2 12 19680 19884,8.

		2			3				dx
1.3.3.											.


								16 x2
		2			2			16 x2
													2	3
	2	3 dx											2	3		3			2
.	2	3 dx						arcsin				x			arcsin	3	arcsin		2	.

			16 x2
	2						4									2		2 12
		2
													2 2
													2 2
1.3.4.		1				x5dx
										.
										.
						x 2
.				1
.							,											«			»:
							,											«			»:

										x5										x 2
																				x 2
																	x4 2x3 4x2 8x 16
												x5 2x4					x4 2x3 4x2 8x 16
														2x4
														2x4			4x3
														2x4			4x3
																	4x3
																	4x3 8x2
																			8x2
																			8x2			16x
																			8x2			16x
																							16x
																							16x			32
																							16x			32

																										32
1	x5dx				1		4		3				2					32							x5		1	x4	1	x3		1		x2	1
1	x5dx				1		4		3				2					32							x5		1	x4	1	x3		1		x2	1
						x		2x		4x				8x 16							dx						2		4				8
	x					x		2x		4x				8x 16					2		dx						2		4				8
1	x	2			1													x	2						5		1	4	1	3		1		2	1
1					1																						1		1			1			1
16x				1	32ln \| x 2 \|						1			2		8	32 32ln 3							46		32 32ln 3 35					1		32ln 3 .
				1							1			2		8								46							1
			1								1
														5	3								15							15


									/12
1.3.5.											sin2 3x dx .

/12

							/12
		.						sin2 3x dx
						/12
		1								1
			sin		sin						.
			sin		sin						.
	12	12		2				2	12	6
					ln 3	3		ex 2 dx.
1.3.6.						3		ex 2 dx.

1	/12		1		sin 6x	/12

		(1 cos6x)dx		x

2	/12		2		6	/12

	0		3		ex 2 dx				9 6ex
		ln 3						ln 3				ln 3	ln 3	ln 3
.										e2x dx 9 dx 6 exdx				e2xdx
	0							0		0			0	0
9(ln 3 0) 6ex		ln 3		e2x		ln 3	9ln 3 6(3 1)				1	(9 1) 9ln 3 16 .



		0			2	0				2
		0			2					2
		/2
		/2
1.3.7.		sin x cos 2x sin 3x dx .
.	0												,
.													,

sin ax cosbx 1 sin(a b)x sin(a b)x 2

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