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Государственный университет — учебно-научно-производственный комплекс (бывш. ОрелГТУ)

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Алгебра и геометрия

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Краткое справочное пособие по школьному курсу математики Определения; Теоремы; Свойства; Формулы; Алгоритмы

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5.1.2. Ceoiicmea

an = at	+ d (n -		1) (Q:>opMyJIa n-ro '1JIeHa);
at	+ an	n =	2a t	+ d (n	- 1)	(<popMyJIa CYMMbI nepasix n
Sn =	2			2	n
'1JIeHOBnporpeccan: Sn = at + ai + ...						+ an)·
llOCJIe,llOBaTeJIbHOCTb				(an)	}lBJI}leTC}I ~pH<pMeTH'IeCKOH nporpec-

cHeH rorztr, H TOJIbKO rorna, xorzra Ka>K,llbIH ee '1JIeH, KpOMe nepsoro

(H nocneznrero, B cnyxae KOHe'lHOH nOCJIe,llOBaTeJIbHOcTH), pasen cpeznrexy a,H<pMeTH'IecKoMyCBOHX COCe,llHHX 'IJIeHOB:

an -	t + an + t		v
an =	')	(xapaKTepHCTH'IeCKOeCBOHCTBO).
	5.2. feOMeTpH'IeCK3J1nporpeccna
		5.2.1. Onpeoenenue
FeOMempU'IeCKOU		npoepeccueu Ha3bIBaeTC}I	nOCJIe,llOBaTeJIbHOCTb

bl' b2 , .•. , bn , ... , Ka>K,llbIH '1JIeHKOTOpOH, KpOMe rrepsoro, nony-raercs nyresr YMHO)I{eHH5l npezrsrztymero '1JICHa ua O,llHO H TO )I{C 'IHCJIO

q :t 0:

bn+ 1 = bn q (q:t 0, b, ::I: 0), rzte q ---- 3HaMCHaTeJIb nporpeccna.

		5.2.2. Ceoiicmea
bn = btqn - t		(diopxyna n-ro '1JICHa).
	bt(qn -	1)
Sn =	q- •	(<popMYJIa CYMMbI nepnstx n '1JICHOB nporpeccaa:

Sn = b, + b2 + b3 + ... + bn)·

ECJIH Iq I< 1 H S = b, + b2 + b3 + ... + bn + ..., TO

S= -- .

1 - q

llOCJIC,llOBaTCJIbHOCTb (b n) }lBJI}lCTC}I rCOMCTpH'ICCKOHnporpeccneii

rorzia H TOJIbKO Torzra, xorzta Ka)I{JJ.bIH ee 'IJIeH, KpOMC nepsoro (H nOCJIC.llHcrO, B CJIy'lac KOHC'IHOH nOCJIC.llOBaTCJIbHOCTH), panCH ITO MO,llYJIIO CpC,llHCMy reOMCTpH'IcCKOMyCBOHX COCC.llHHX 'IJICIIon:

Ibn I=..Jbn _ t bn + t . I1JIH, 'ITOTO )KC caMOC, b~ = bn _ t bn + r-

....~\.,.. '

'II

'·1'

iii,

TPHfOHOMETPIIH

1. TpHTOHOMeTplflleCKHe <l>yHKIJ;HH

1.1. 'lHCJlOBaJIoKp~OCTh

1.1. 1. Onpedenenue

Flyers zrana OKPY)KHOCTb panayca 1. nOCTaBHM B COOTBeTCTBHC Ka.>K,llOMY .lleHCTBHTeJIbHoMy 'IHCJIYt TO'lKyOKPY)KHOCTH no CJIC,llylO-

nrexy rrpasany:

ECJIH t = 0, TO cMy COOTBeTCTByeT TO'lKaA - npaBbIH KOHCU rOpH30HTaJIbHOrO .llHaMeTpa.

ECJIH t > 0, TO, OTnpaBJI}lJlCb H3 TO'lKHA, onaurex no OKPY)KHOCTH B aanpaaneaaa nporas '1aCOBOHCTpe.nKH nyrs .llJIHHOH t; KOHeu; aroro nYTH H 6Y.lleT HCKOMOH TO'lKOHM(t).

ECJIH t < 0, TO, OTnpaBJI}lJlCb H3 TO'lKHA, ormurex no OKP)')KHOCTH B HanpaBJIeHHH no '1aCOBOH CTpe1IKe nyrs .llJIHHOH I t I; KOHeu; aroro nYTH H 6Y,lleT HCKOMOH TO'lKOHM(t).

E,llHHH'IHMOKPY)KHOCTb C YCTaHOBJIeHHblM COOTBeTCTBHCM Ha3bI-

saercs '1uCJJo8ou 01CpYJKHOCm&lO.

Ka.>K,llOMY .lleHCTBHTeJIbHOMy 'IHCJIycooraercrsyer C,llHHCTBeHHM

TO'lKaOKPY)l{HOCTH.

I I I A

ECJIH	TO'lKa M COOTBeTCTByeT '1HCJIy t, TO OHa COOTBeTCTBYCT
JIlo60My '1HCJIYnazta t + 21tk,			rzre 21t - ,llJIHHa C.llHHH'IHOHOKpy)KIIO-
CTH, a	k -	UCJIOC '1HCJIO,	noxassmaiouiee KOJIH'ICCTBO nOJIHbIX

06XO,llOB OKpy)KHOCTH B ry HJIH HHylO CTOpOHy.

~.	1!)

1.1.2. J(6a OCH08HblX uaxema IIUCJl060U OKpYJlCHOcmu:

	-1<2			"2"
			511
			6"
"I	I	10	xl	I	10
	31&	T	ss:	3r.	ft
	T		3	T	3
		1.2. TPHroBOMeTpHlIeCKHe 4>YHKIUlH
		1.2.1.	Onpeoenenue

sin t - opaaaara TOqKH M(t) qHCJIOBOH OKPY)I{HOCTH; cos t - aocuacca TOqKH M(t).

	tg t = sin	t
A	cos	t .
A	ctg t = cos t
J(	ctg t = cos t
	sin	t .

1.2.2. 3HaKu no «emeepmsu

sin t

C05 t

tg t ctg t

		1.2.3. Ceoiicmea
sin	t, tg t, ctg t -	nesemsre cPyHKUHH, cos t - qeTH3jI.
21t - rtepaozi sin t, cos t,
1t -	nepaozi tg t,	ctg t.

'Ji

		1.2.4.	OCH06Hble 3HalleHUJI

				Apryxeur
<l>yHKUHSI
<l>yHKUHSI	0	1t	1t		1t		1t	1t	31t
		1t	1t		1t		1t		31t
		-	-		3		2		2
		6	4		3		2		2
sin t	0	1	..f2		-./3'	I	.	0	- 1
		1	..f2		-./3'	I
		-	"2		2
		2	"2		2

cos t		..f3	..f2		1			- 1
	1	2	2		-		0		0
	1				-		0		0
					2
tg t	0	..f3	1		..f3		----	0	----
tg t	0	3	1		..f3		----	0	----
		3
ctg t	---	..f3	1		{3		0	----	0
					{3
					3
					3
1.3. 06pamble TPHroHoMeTPHlIeCKHe 4>YHK~HH
		1.3.1. Onpeoenenus
arcsin m	--- 3TO ztyra, CHHyC KOTOpOH pasen m H KOTOpa}! 3aKJIIO-
xeaa B 3aMKHYToM npoxexcynce OT -					1t	1t
xeaa B 3aMKHYToM npoxexcynce OT -					"2 no "2 :
		.	.			1t		1t
	Y = arcsin m <=> SIll Y = m,					-"2 ~ y ~ 2'

arccos m -- 3TO ztyra, KOCHHyC KOTOpOH pasen m H KOTOp3jl

3aKJIIOqeHa B 3aMKHYTOM rrpoxescyrxe OT 0		no 1t	:
y = arccos m <=> cos y = m,		0 ~ y ~ 1t.
arctg m ----	3TO ztyra, TaHreHC KOTOpOH paaen			m H	KOTOpa}!
3aKJIIOQeHa B OTKpbITOM npoxeacyrxe OT -		1t	1t
3aKJIIOQeHa B OTKpbITOM npoxeacyrxe OT -		2 no	2 :
Y		1t	1t
Y	= arctg m <=> tg Y = m, -2 < y < 2 .
arcctg m ---	3TO nyra, KOTaHreHC KOTOpOH		pasen	m H	KOTOpa}!

3aKJIIOQeHa B OTKpbITOM npoxexcyrxe OT 0 no 1t:

y = arcctg m <=> ctg Y = m, 0 < y < 1t.

1.3.2. OCHo8Hbie coomnotuenus

	arcsin (- x) = - arcsin x.
	arccos		(- x) = 1t			- arccos x.
	aretg		(- x)		=- arctg x.
	arcctg (- x)				= 1t - arcetg x.
						1.3.3. OCHo8Hbie 3HalteHUR

										AprYMeHT
	<DYHKUIUI
	<DYHKUIUI						1			..ff			..f3
					0		1			..ff			..f3	1
					0		-				2		2	1
							2				2		2

	arcsin x							1t			1t		1t	1t
	arcsin x				0		6				4		-	-
							6				4		3	2
	arccos x				1t			1t			1t		1t	0
					1t			1t			1t		1t
					-		-				-		-
					2		3				4		6

										Apryxenr
	<DYHKUIUI
	<DYHKUIUI								..f3
									..f3					..f3
						0			3-			1		..f3

	arctg x					0			1t				1t	1t
									1t				1t	1t
									-			-		-
									6			4		3
	arcctg x					1t			1t				1t	1t
						-			-			-		-
						-			-			-		-
						2			3			4		6
.-
1.3.4.		OCHo8Hbie rPOPMYflbl, C8Jl3bl8alOUI,ue mpueououempusecxue U
				otipamnue mpueonouempusecsue rPYHKl{UU
	sin (arcsin x)					= cos (arccos x)				= x,	(-1	~ x ~ 1).
	tg	(arctg x)				=etg (arcctg x)				=x,	(r *- 0).
	sin	(arccos .r)				= cos (arcsin x) =-V1-x2 ,						(-1 ~ x ~ 1).
	tg	(arcctg .r)				=ctg (arctg				1
	tg	(arcctg .r)				=ctg (arctg			x) =-, (r *- 0).
										x
22

f·

1.4. rpa~HKH TpHfoHoMeTPH'IeCKHX~YHKU.Hit

.__._

.__..

1.5. fpa$HKH ofipannax lpHrOHOMelpH'IeCKHX$YHKIlHA

y=arcsinJ( y=arCC05X

		!J
		-l	.-l
			1l
			I
			I
			I
			I
			I
	-1	0 1 X
	y=aretgx
	.!Ii"
	'2
		x
___.	!7L	_
___.	:"'2	_
	y = arc ctq x
	!I
	1<
------	~------

L/j

j~~

',~

I'~

,\t

-Iis

			2.		<I>OpMyJlbI TpHfoHoMeTpHH
		2.1. <I>0PMYJIbI, CBB3b1BaIOInHe 4>YHKIlHH
					onnoro H TOro )ICe apryxesrra
sin2x + cos2x = 1.									2	1	•
sin2x + cos2x = 1.								1 + ctg x - siIlx
	2 __1 .							tg x ctg x		= 1.
1 + tg x -			cos2X					tg x ctg x		= 1.
1 + tg x -			cos2X
2.2. <!>OpMyJlbl, CBB3b1BalOUJ,He 4>YHKIlHH apryaenroa,
		H3 KOTOPblX O,ZUIH BABoe 60Jlbme Ap.yroro
sin 2x =2 sin x cos x .
cos 2x = cos2x - sin2x.
tg2x=		2 tgx
tg2x=		1 -	tg	2'
		1 -	tg		X
2	=	1 -	cos 2x			2		1 + cos 2x		(<l>OPMYJIbI nonaaceaas
sin x	=		2		,cos x =				2	(<l>OPMYJIbI nonaaceaas
creneaa).
1 ± sin 2x =(cos x ± sin X)2.
		x	= U,			.	2u			1 - u2	(YHHBepCaJIbHM
ECJIH tg -2			= U,		TO Sill X =			2' cos X =		?	(YHHBepCaJIbHM
						1+u				1+u
rrO,ltCTaHOBKa) .
			2.3. <I>OPMYJlbI CJlO)ICeHHB apryaearoa
sin (ex. ± ~) =sin ex.						cos ~ ± cos ex.			sin ~.
cos (ex. ± ~) = cos ex. cos ~ -+							sin ex.		sin ~.
t (ex. ± A) = tg ex. ± tg ~ .
g		p	1+tgex.tg~
											2S

;;Ii

2.4. <DOPMYJIhl npeo6pa30BaHHJlcyMM B npoH3BeAeHHJI

.	+. A		=	2'ex±p			ex:+: p
SIll ex -		SIll ...			Sill --2- cos -2- .
					ex+P		ex-p
cos ex + cos p =2 cos - 2 - cos - 2 - .
cos ex - cos P=					ex+P		ex-p
					-2 sin - 2 - sin - 2 - .
tg ex ± tg p=			sin (ex ± A)
					.....
			COS ex COS p
A sin t		+ B cos			t = -YA2 + B2		sin (t + <p), rzie <p - BCIlOMOraTeJIb-
u						A
HbIH yrOJI,		IlpHqeM cos <p = -YA2 ;: B2 .
	2.5. <DOpMY.7lbl npeo6pa30BaHHJI npoH3BeAeHHlt B CYMMbI
sin ex cos p = ~ (sin (ex -						P) + sin (ex + P».
sin ex sin P =~ (cos (ex -						P) -	cos (ex + P))·
			1		(cos (ex -	P) + cos (ex + P)).
cos ex cos P ="2

2.6. <DOPMYJlbI npHBeAeHHJI

3TO cPOpMyJIbI, C IlOMOlllbIO KOTOpbIX TpHrOHOMeTpHqeCKM cPYHK-

uas OT apryxenra BH,ua 2 ± ex npeoripaayercs B TpHrOHOMeTpHqeC-

KyIO cPYHKllmo OT apryxenra ex.

Ilpaeuno iJJlR 3anOMUHaHUJI tjJoPMYIl npuseoenus:

1. .llJI}I apryxetrros, OTCqHTbIBaeMblX OT ropasoaransaoro ,uHaMeTpa (n ± ex, 2n - ex), Ha3BaHHe cPYHKllHH He MeH}leTC}I.

2 . .ll.JI}I apryxenroa, OTCqHTbIBaeMblX OT BepTHKaJIbHOrO naaxerpa

n	3n	± ex),	~.	(	CHHyC na KOCHHyC,
(2 ±	ex, 2	± ex),	Ha3BaHHe 't'YHKllHHMeH}leTC}I		CHHyC na KOCHHyC,

KOCHHyC na CHHyC, TaHreHC na KOTaHreHC, KOTaHreHC na ranrenc).

3. Ilepen IlOJIyqeHHOH cPYHKllHeH CTaBHTC}I TOT 3HaK, KOTOpbIH HMeJIa 6bI IlpHBO,uHMM cPYHKllH}I B TOH xeraeprn, B KOTOpOH JIe:>KHT

nn	± ex,	n
apryxenr 2	± ex,	eCJIH 0 < ex < 2'

2.1. Ilpocremnae TpHroHoMeTPH'IeCKHeypaBHeHUJI

sin x = a, Ia Is 1			<=> x = (- 1)" arcsin a + nn.
cos x = a, Ia I~ 1			<=> x = ± arccos a + 2nn.
tg x = a	<=>	x = arctg a + nn.
ctg x = a	<=>	x = arcctg a + nn, n E Z.
'Iacmnue cnyuau:
sin x = 0	<=>	x = nn.
cos x = 0	<=>		n
		x ="2 + nn.
sin x = 1	<=>		n
		x = 2 + 2nn.
cos x = 1	<=>	x = 2nn.
sin x = -1	<=>		n
		x =-"2 + 2nn.
cos x = -1	<=>	x = n + 2nn.

3JIEMEHThI ,lJ.lI<I><I>EPEHUlIAJIbHOrO HClIHCJIEHHSI

	t. npOH3BO.llHaS
	1.1. Onpeneaeaae npoH3BOAHoA
f'(x o ) = lim	.M.,
~x--+o	tu

rzte tif = f(x) - f(x o) (npapauicnae <PYHKUHH); ti x =x - X o (npn- pameaae apryxeura).

1.2. (J)OpMyJlbl AR4J~epeHUHpoBaHHJI

(c)' =O.

(kx + b)' = k.

(x')' =rx'- 1 •

(if)' = ifIn a.

(ex)' =e',

(In x)' = 1. x

1 (logax)' =-1 .

x n a

(sin x)' =cos x. (cos x)' = - sin x.

(tg x)' =_1 .
cos2	x
(etg x)' =	1_.
sin2 x
(arcsin x)' =	I	1	') (-1 < x < 1).
	1-x
(arccos x)' =-		1		(-1 < x < 1).
(arccos x)' =-		I	..	(-1 < x < 1).
		1-x
(arctg x)' =	1	~.
1+x
(arcctg x)' =__1			~.

1+x

1.3. Ilpasaaa 4H~~epeallHpoBaIBlJl

(u ± v)' =u' ± v'.

(ku)' =ku' (k -		'IHCJIO).
(uv)' =u'v + uv'.
~) _ u'v	~ uv'
(V -	v2	•

(u(v(x)))' =u'(v(x»v'(x) (rrpoH3BO,llHCUI CJIO)l{HOH <PYHKUHH).

1.4. feoMeTpH'IecKHACMhlCJI npoH3BO,lUlOft

fta) = tg ex = k, rae k - yrJlOBOH K03cP<PHUHeHT xacarensaoa,

y:f(x)

1.5. YpaBHeHHe KaC3TeJlbHoft

y = f(a) + ('(a) (x - a).

a cnpasa nonoxorrensna,

2. Hccxenonaaae <l>YHKilHH C nOMODI,bIO

npoH3Bo,ZJ;HOH

2.1. lIcCJleAOBaHHe H3 MOHOTOHHOCTh

EClIH f{x) > 0 na npoxeacyrxe X, TO <PYHKUH~ Y =f (X) B03paCTaCT

HaX.

EClIH f{x) < 0 na npoxexcynce X, TO <PYHKUH~ Y = f (X) yrismaer

HaX.

2.2. lIcCJle,lJ,OBaHHe H3 3Kc'l'peMYM

2.2.1. Onpeoenenue

EClIH y TO'-lKH X o cymecrnyer OKpCCTHOCTb, B KOTOpOH <PYHKUH~

Y = f (x) onpezte.nena H f (x) s f (xo)' TO Xo Ha3bIBalOT mO'lKOU MaK-

cUMyMa 4JYHK~UU;nHIIIYT Ymax =f (X O).

EClIH Y TO'-lKH X o cymecrsyer OKpCCTHOCTb, B KOTOpOH <PYHKUH~

Y =f (X) onpezienena H f (X) ;::: f (xo), TO Xo Ha3bIBalOT mO'lKOU MU-

HUMyMa 4JYHK~UU;nHIIIYT Ymin = f (x o)·

2.2.2. AIlzoPUmM omblCKaHUJI Yrmx, Yrrin aM rPYHK~UUY =f (x)

1. HaH,llHTC 0611aCTb OnpC,llClICHH~<pyHKUHH.

2. HaH,llHTC Y' ::;: (x).

3. HaH,llHTC TO'-lKH,B KOTOpbIX (x) = 0 HlIH (x) HC cymecrnyer , H BbI6cpHTC H3 HHX TC, '-ITO npaaaanescar 0611aCTH OnpC,llClICHH~

<pyHKUHH.

4. OTMCTbTC BbI6paHHbIc TO'-lKHna '-IHClIOBOHnp~MOHH OnpC,llClIHTC

3HaKH Y' ClICBa H cnpasa OT Ka)f(,llOH H3 OTMC'-ICHHbIXTO'-lCK.

S. C,llClIaHTC BbIBO,llbI: CClIH npOH3BO,llHM Y' ClICBa OT OTMC'-ICIIHOH TO'-lKH X o orpauarensna, TO X o ---- TO'-lKa

MHHHMyMa H f (x o) = Ymin; CClIH npOH3BO,llHM Y' cnesa OT OTMC'-ICHHOH TO'-lKH X o nonoaorrensna, a cnpasa OTpHuaTClIbHa, TO X o ---- TO'-lKa

MaKcHMyMa H f(x o) = Ymax '

2.3. OTblCK3HHe H3H60Jlbwero H H3HMeHbwero 3H3lJeHuit

uenpepsmaoa 4>YHKD;HH H3 npoMe*ytKe

2.3.1. Onpeoeneuue

rOBOp~T, '-ITO <PYHKUHSI Y = f (x) ztocruraer na npoMClKyTKC X

csoero HaUOOJl."zuezo (HaUMeH"zuezo) 3Ha'leHUJl, CClIH cymccrnyer

TO'-lKaX o E X raxas, '-ITO,lllla BCCX x E X asmonuaercs HcpanC::HCTBO f (x) s f (xo) (f (x) ;::: f (xo)); nnuryr Y HaH6 = f (xo) (YHaItM= f (.xo» .

Henpepsmuaa <pyHKUHa na OTpC3KC scerna ztoctaraer cnoero

nanrio.nstuero H HaHMCHbIllcro 3Ha'-lCHHa.

2.3.2. Aneopumu omblCKaHUJI YHalt6, YHaHM aM rPYHK~UUY = f (X), nenpepuenoii ua ompesxe [a,b]

1. HaH,llHTC !'(x).

2. HaH,llHTC TO'IKH,B KOTOpbIX f'(x) =0 HlIH (x) He CYUlCCTUyCT, H BbI6cpHTC H3 HHX re, 'ITOnexcar snyrpa OTpe3Ka [a,b].

3. Cocransre Ta611H11Y 3ual.ICHHH <pyHKUHH, xyzra BKlIIO'IHTCTO'IKH a, b H TO'-lKH, HaH,llCHHblC na mare 2.

4. Ha naiiztetnrstx 3Ha'-lCHHHqlyHKUHH BbI6cpHTC HaH6ollhl1JCe (:'ITO oyzter Y Hillt6) H HaHMCHbllICC (iJTO 6Y,llCT Y HaltM).

2.3.3. CnYlfau nesauxnymoeo npouescymxa

Henpcpsmnas qlYHKUH~ na HC3aMKHyroM npoxeacyrxe MO)l(CT HMeTb H MO>KCT IIC HMeTb Y Halt6'YHaHM•

Ilpocmeiauue cnynau:

EClIH nenpepsranas cPYHKUHSI Y =f (r) HfolCeT B nponeacyrxe X TOllbKO O,llHY TO'-lKysxcrpeayxa Xo H CCllH X o --- TOtIKa MaKcHMyMa,

TO f (xo) = Y HaH6'

EClIH ncnpepsranaa cPyHKuHa Y = f(x) HMCeT B npoueacyrxe X

TOllbKO O,llHY TO'IKysxcrpeayxa Xo H CCllH X o -- TO'IKaMHHHtvlyMa, TO f(xo) = YHaHM.

3JIEMEHTbI MHTEfPAJIbHOfO

MCQMCJIEHMH

1.TIepBQ06pa3HaH

1.1.OnpeAeJleHHe nepBoo6pa3HoA

ECJIH JlJI}I	JII060ro x H3 npoxexyrxa X BbITIOJIH}lerC}I paaencrso
F'(x) ={(x),	TO <PYHKUH}I F (X) Ha3bIBaerC}I nep8oo6pa3HOU JlJI}I
{ (X) aa npoxescyrxe X.
1.2. Ilpasaxa Bbl1JH«JleHHJI nepBoo6pa3HoA
ECJIH F (x) -- nepBo06pa3Ha}{ JlJI}I {(x), a H (x) -- nepaootipaa-

nas JlJI}I h (x) na npoaescyrxe X, TO F (x) + H (x) --- nepaootipaanas

JlJI}I {(x) + h (x) na npovexyrxe X.

ECJIH F (x)

- nepsooopaanas JlJI}I {(x)

na npoxescyrxe X,

kF (x)

---

nepBo06pa3Ha}{ JlJI}I k{(x)

npoxexyrxe X (k

----

npOH3BOJIbHOe JleikTBHTeJIbHOe qHCJIO).

ECJIH F(x)

---- nepBo06pa3Ha}l

JlJI}I

((x)

na npoxexcyrxe X,

na nponeacyrxc X.

- F(ax + b) -

nepBo06pa3Ha}{ JlJI}I {(ax + b)

1.3. <DOpMyJlbl Bbl1JHCJleHHJI nepBoo6pa3HoA F (x)

AJIJI cPYHKIJ,HH f

(x)

{(x)

I t

Ix' (r ~-

t) I;(x> 0)

I sin

I cos x

I Si~2x I co~x

F(x)

I x

oX'" + 1

Inx

I - cos x

I sin x

1- ctg x I tg x

r + 1

ax (a > 0, a e 1)

f(x)

F(x) I eX

arctg x

arcsin x

In a

2.Heonpe,ZJ;eJIeHHbIU UHTerpaJI

2.1.Onpenexeane aeonpeaeaeaaoro mrrerpaza

ECJIH F(x) --- nepaooripaanaa JlJI}I {(x) aa nposrexcyrxe X, TO MHO)l{eCTBO BCeX nepaooripaansrx JlJI}I {(x) HMeer BHJl {F(x) + C}, rne

C -- JII060e JleikTBHTeJIbHOe qHCJIO. 3TO MHO)l{eCTBO Ha3bIBalOT

HeonpeiJeAeHH&IM UHmetpaAOM c]JYH1Ct4UU f<x) H 0603HaqalOT

f {(x) dx (QHTaeTC}I "HHTerpaJI 3<p OT HKC Jl3 HKC"):

f((x) d x = F(x) + C.

2.2.Ilpaaaxa HHTerpHpooaHHSI

f (((x) + h(x» dx =f ((x) dx + f h(x) dx.

fk{(x) dx = kf {(x) dx.

2.3.<DOPMYJlbl narerpnpoaaana

f dx =x + C.	'+1
f dx =x + C.	f x' fix = ~1 + C (r "# - 1).
	r+

f	_dx =In Ix 1+ C.
	x	-
	dx
f cos-.2- x		=tg x + C.

f-:__ = - ctg x + C.

SIll X

f sin x dx =- cos x + C.

f ~os x dx = sin x + C.

f eX dx = eX + C.

fr? dx =			ax
fr? dx =			_I_ _ + C (a > 0, a e 1).
			n a
f	dx	= arctg x + C.
	----:-2	= arctg x + C.
	1+x
	dx		.		C
f A \11I - x 2			=arCSlll	x +		.

:i:i

3.Onpeneaennsrit HHTerpaJI

3.1.<!>opMyJla HbIOTOHa-JIeit6HHIla

ECJIH F(x) ---- nepsooripaanaa j:{JI5I f(x) na rrpostexcyrxe X H eCJIH a, b ---- TO'IKHH3 aroro nposrexcyrxa X, TO

f f(x) d x =F(b) - F(a) (q)OpMyJIa HbIOTOtIa-JleH61IHua),

II
b
me f f(x) dx ----		onpeaeJtellllbtU uumeepas; a, b ---- npenensr HHTer-
II
pHpOBaIIH5I; f(x)		---- rtozu.urrerpant.uaa epYHKUH5I.
	3.2. Ceoncraa onpeaexeaaoro mrrerpana
b			b		b
f ( f(x) + hex) ) dx =f I<x) d x + f hex) d x,
II			a		/I
b		b
fkf(x) dx = k fI<x) dx,
II		a
b		/I
f f(x) d x = -		f I<x) d x,
II		b
		b		c	b
ECJII1 a < c < b, TO f			f(x) dx = f		f(x) dx + f	f(x) d x (aJIJII1TI1BHOe
		II		a	c
CBOHCTBO mrrerpana).
				b
ECJIH f(x)	~ 0 na [a;		b],	TO f f(x) dx ~ O.
				/I
				b	b
ECJIH f(x)	~ g(x) ria		[a;	b], TO f	f(x) fix ~ f	g(x) d x.
				/I	a

3.3. Bsrsucxenae nnouianen nJlOCKUX <j>uryp

C noxoutsro mrrerpana

ECJIH epHrypa D npeJICTaBJI5IeT C060H qaCTb nJIOCKOCTH xOy, orpana-rennyro np5IMbIMH X = a, x = b (a < b) H rpaepHKaMH aenpe-

pbIBHbIX ua OTpe3Ke [a; b]			epyUKUHH Y =f(x), y :::; h(x)	TaKHX,	'ITO
JIJI5I .morioro	x	113 [a; b] BbillOJIH5IeTCH nepaseucrno h(x) :5 f(x),			TO
nJIOmaJIb	S	qJHrypbI	D Bbl'iHCJI5IeTC5I no	epopMyJIe:

S =f(((x) - hex)) dx .

b x

')r:

,J. )

.._. __ •.

-·--~==~7. ," _.~_

=:-==--<I-=C'.

IIJIAHMMETPMH

1.TpeyroJlbHHKH

1.1.OOOaHa'iemUI

A, B, C - yrnsr; a, b, c - CTOpOHbI; he -- BbICOTa, nposeneanas

K CTOpOHC c; me -- MC.llHaHa, npOBC,llCHHa51 K cropone c; S ---

rrnoutans: R	-	pazrayc		onHcaHHoH			OKpy)KHOCTH;	r --- paznryc
BnHcaHHOH OKpy)KHOCTH; p -					IIOJIyrrcpHMerp.
		1.2. PaBHOCropOHHHii -rpeyrOJIbHHK
A =B = C =60';					a ...g. S = a2...J3
a...J3		a ...J3			a ...g. S = a2...J3
h = -- ' R		= -- ' r =			6	'	4
2	'		3	'	6	'	4
1.3. npJlMoyrOJlbHblii TpeyrOJIbHHK (C= 90·)
		1.3.1. Mempusecxue coomnotuenus
c				a2	+ b2 = c2 (reopexa Ilarparopa):
c				a2 = ca';
				a2 = ca';
				2
~ b2					=cb';
A h' H	a'	8		h	= db';
				h = a b.
					c
				1.3.2. llAow,aab
					S = a2b .
1.3.3. c[JOPMYJlbl aM 6blttUCJleHWI paaUYC06 R H r
				c		a+b-c
			R=2' r =				2
1.3.4. Coomnouienus MeJICi)y cmOpOHaMU U YZ/laMU
	sin A = !!-				b		a	b
	sin A = !!-		cos A =-,			tg A =-b' ctg A		=-.
		c'			c			a
36

1.4. npoH3BOJIbHblii TpeyroJIbHHK

1.4.1. 'Iemupe sauesamensnue mosocu 6 mpeYZOJlbHUKe

Mezutanta TpcyrOJIbHHKa nepeceicarorcs B O,llHOH TOtIKe H ,lleJI5ITC5I B HCH B OTHOWCHHH 2 : 1, CtIHTa51 01'BepWHHbI. ::ha TOtIKa Ha3bIBaCTC5I

ueumpo» muscecmu.

Bb1COTb1 rpeyrom.aaxa nepecexaiorca B O,llHOH TOtIKe. 3Ta TOtIKa Ha3bIBaCTC5I opmouenmpox;

Bnccexrpncsr TpeyrOJIbHHKa nepecexatorcs B O,llHOH TOtIKe. 3Ta

TOtIKa HBJI5ICTC5I Z1eHmpoM 8nucaHHou oKpyJKHOcmu.

TIcpnCH,llHKyJI5Ipbl, npOBC,lleHHbIC K croponan TpcyrOJIbHHKa sepes

HX cepezram.i, nepccexaiorcs B O,llHOH TOtIKC. 3Ta TOtIKa 5IBJI5IerC5I

Z1eHmpoM onucaunou oKpyJKHOcmu.

1.4.2. Cpeons» JlUlIWl mpeyeonsnuxa

TIapaJIJICJIblla ocuosaumo.

PaBHa nOJIOBHlIC OCHOBaHH5I.

lleJIHT nOnOJIaM JIlO60H oTpe30K, COe,llHH5IIOLUHH BCpWHHy rpey-

rOJIbHHKa C KaKOH-JIH60 TOtIKOH OCHOBaHH5I.

1.'1.3.C60ucm60 tiuccexmpucu mpeyeonsnuxa

						TIYCTb B TpeyrOJIbHHKC ABC rtpose-
						zien a	6ncceKT p aca BD. To r n a
						AB	AD
						Be =	DC (6HCCcKTpHca ,llCJIHT OCHO-
~ BalIHC							na 'IaCTH, nporropunouansnsre
A	D					C 60KOBblM croponax).
						1.4.4. Onpeoenenue euoa mpeyzons-
						HUKa no eeo cmoponau
TIYCTb c ---- HaH60JIbWaH H3 TpCX CTOpOH TpeyrOJIbHHKa.
2	< a	2	+	b2			u
ECJIH C	< a		+	b2	,TO TpeyrOJIbHHK oCTpoyrOJIbHbIH.
2	= a	2	+	b2			u
ECJIH C	= a		+	b'>	,TO TpcyrOJIbHHK np}{l'loyrOJIbHbIH.
2	> a	2	+	b'>
ECJIH C	> a		+	-, TO TpcyrOJIblIHK rynoyronunsnr .
S =1c ltc,						1.4.5. flJlOU{aab
S =1c ltc,		S =P r,				S =~ a b sin C,
S = ...Jp (p -		(1)		(p -		b) tp - c) (q)OpMyJIa Fepoua).
							37

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