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‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï	31

58. R arctg2 x dx ; p 1+x2

59. R 3parcsinx dx;

1 x2

60. R 1 2sin2 x dx; cos x

61.R 1+sin2sin2 x x dx;

62.R esinx cosx dx;

R etgx

63.pcos2 x dx;

64.R p5 x3 8x2 dx;

65.R 4 1 6x5 x4 dx;

p

66. R 2pxx dx;

1

67. R 3xx2dx ;

68. R p3x+5 dx;

4x+1

69. R p5x 6 dx;

1 3x

70. R p2 4x dx;

7x 1

Rearctgx

71.1+x2 dx;

Rsin 12

72.x3x dx;

73.R 41 3x dx;

74.R arccosx dx; p

1 x2

75. R e tgx sec2 x dx;

76. R arcsinx+x dx; p

1 x2

77.R x2dx16;

78.R x2dx+4;

79.R p4dxx2 ;

80.R p4+dxx2 ;

81.R pxdx2 3;

82.R x2dx 5;

83.R x2dx+3;

84.R 2dxx2 ;

85.R 4xdx2+5;

86.R p25dx4x2 ;

87.R p3+2dx x2 ;

88.R 9xdx2 1;

89.R p5dx3x2 ;

90.R 3 dx5x2 ;

32	‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï

91.R p9dxx2 5;

92.R px2dxx4 ;

R

x3 dx

94.

pe5 dxe2x ;

93.

px8

3;

R

x

95.

sin2x dx

;

5 cos2 2x

96.

R

2xx2

43 dx;

97.

R

px2

+1 dx;

R

x+1

98.

px1+1x2

dx;

99.

R

p5ee2xdx4;

100.R

x

p3cos2 5x

2;

R

cos5x dx

x

101.

sin 3 dx

;

4cos2 x +9

R

3

102.

x4 dx

;

10

R

p4 x

103.R xx146 dx+5;

104.R ee 2xxdx+2;

105.R x2+4dxx+5;

106.	R		dx			;
		2
		x 6x+13
107.		p	dx				;
			x2+2x+3
108.	R	p4x x2 ;
	R		dx
109.		p3 2x x2 ;
	R		dx
110.		p2+3x 2x2;
	R		dx
111.		3x2 2x 1;
	R		dx
112.		x22x2x+5 dx;
		+1
	R		x 1
114.
		(1+x2+xx)dx1			;
113.	R	x25+3x+3 dx;
115.	R		2 x
		p5 4x x2
	R	x2+4x+29 dx;
	R	p4x2+4x+3
116.	R		3x 2					dx;
118.		p65x x2 5 dx;
117.			1 2x						dx;
	R		x+11
		p
120.			6x x2
	R	p3x+2x2 dx;
119.		1 3x			dx;
	R		3+x
121.		px42x+8x+7 dx;
	R	+11
122.		x27x6x1+1 dx;
	R

‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï	33

123.	R	x3	dx;
		x 22
		x +5		dx;
124.		3x+1
	R	4
125.	R	x		dx (a 6= 0);
		x2+a2
		x		4
126.	R	(x 2)(x 3)			dx;

127.R (x2+1)3 dx; x x

128.R 32xx+11 dx;

129.	R	2x2			1	dx;
		2
130.		x4x 23x3				dx;
		x		x+1
131.	R x32x			6x2+10 dx;
	R		3		x		2

R(x3 2x)dx ;

132.x2 8x+7

		3x2+1
133.					dx;
		x2	x+1
	R		3+x
135.		x4x 33x2			dx;
134.	R	x2	+7x+13 dx;
136.	R x2x+8x				7 dx;
			2
	R		+3x

R

137.lnx dx;

138.	R x lnx dx;
139.	R	x ln(3x + 2)dx;
140.		(x2 + 3x + 2)lnx dx;
141.	R	xe x dx;
142.		xe		5x dx
	R						x ;
143.	R	x3e x dx;
144.	R	x2e					2 dx;
145.	R	(2x + 3)e2x dx;
146.	R	x cosx dx;
	R
147.	R x sinx dx;
148.	R		x + 1)cos3x dx;
149.		(x2 cosx dx;
150.	R	x cos2 x dx;
151.	R		x dx		;
			sin2 x
	R		x
152.	R		cos2 x			dx;

R

153.arctgx dx;

154.	R	arcsinx dx;
155.	R	x arctgx dx;
156.	R	x arcctg(1 x)dx;
		arcsinx
157.		p		dx;
			1+x
	R
158.	R	arctgp7x					1dx;

34

‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï

159.

R

x ln 1+x dx;

1 x

160.

ex sinx dx;

161.

R

ex cosx dx;

162.

R

e2x cos3x x dx;

163.

R

ex sin 2 dx;

R

px dx;

164.

R

ln2 x dx;

165.

R

ln5

x

166.

ln(x2 + 2)dx;

167.

R

cos(lnx)dx;

168.

R

x cosx dx

;

sin3 x

169.

R

x tg2 x dx;

170.

R

arctgx

dx;

x2

R

arcsinp

171.

x

dx;

p

x

R

x+2)

172.

ln(

dx;

x2

arctgp

R

x

x px dx;

173.

R

p

x

dx;

R

lnx

175.

p7

x2 dx;

174.

R

3

p

176.

x2

5dx;

R

p

dx;

177.

3 x2

R

p

dx;

178.

x2 + 2

R

p

dx;

179.

2 3x2

R

p

180.

2x2

1dx;

R

p

dx;

181.

6x

x2

R

p

182.

x2

4x dx;

183.

R

px2 + 5x + 4dx;

R

p

dx;

184.

3 2x x2

R

p

dx;

185.

5 + 4x x2

R

p

186.

2x

x2 dx;

188.

R

expe2x

+ 3dx;

187.

R

sinxp2 3cos2 x dx;

190.

R e 2 p4p

ex dx;

189.

R

cosx x sin2 x + 3dx;

R

p

191.ln2 x + 1 dx ;

R p x

192.(2x 1) 3x x2 dx;

R p

193.(x + 3) 5x + 2x2 dx;R

‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï

35

R p

194.(x 1) 6x x2 dx;

195.R sin2 x dx;

196.R cos2 x dx;

197.	R	sin2 mx dx								(m 6= 0);
198.		cos2 mx dx								(m = 0);
199.	R	sin3 x dx;								6
200.	R	cos3 x dx;
201.	R	cos4 x dx;
202.	R	sin5 x dx;
203.	R	cos2 x sin2 x dx;
204.	R	sin3 x4						cos3 x4 dx;
205.	R	sin2 x						cos4 x dx;
206.	R	cos2 x					sin4 x dx;
207.	R	sin3 x cos2 x dx;
208.	R	cos7 x dx;
209.	R	sin4 x cos4 x dx;
210.	R	cos3 x sin5 x dx;
211.	R	sin4 x2 dx;
212.	R	(1+ 2cosx)2 dx;
213.	R	cos5 x dx;
214.	R		dx		;
			sin2x
215.	R		dx	;
			cos x3
216.	R		dx		;
			sin9x
217.	R		dx			;
			cos5x
218.	R		x+cosx						dx;
			sinsin2x
	R

219.	R	sin3x cosx dx;
220.	R	sin3x sin5x dx;
221.	R	sinnx sinmx dx (m + n 6= 0, m n 6= 0);
222.	R	sin3x sinx dx;
223.	R	sin 5x 4	cosx dx;

224.sin x3 cos 23x dx;

225.R cossin3 2xxdx ;

226.R cossin32 xx dx;

227.R ctg3 x dx;

228.R cossin53 xx dx;

229.R cossin53 xx dx;

230.R tg4 x dx;R

36	‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï

231.R sindx4 x ;

232.R 1+3cosdx 2 x ;

233.R 5+3cosdx x ;

234.	R	dx	;
		3sinx+4cosx

235.3+cosdx x ;

236.R tg5 x dx;R

237.

R

dx

;

2sinx+sin2x

238.

1+cosx

dx;

sin4 x

239.

R

dx

;

sinx cosx

240.

R

dx

;

sinx+cosx

241.

R

dx

;

3sin2 x+5cos2 x

242.

R

dx

;

sin2 x+3sinx cosx

cos2 x

R

dx

243.

;

sin2 x 5sinx cosx

244.

R

dx

;

8

4sinx+7cosx

245.

R

dx

;

(sinx+cosx)2

246.

R

sinx dx

(b = 0);

b2+cos2 x

247.

R

cos5 x

dx;

6

sin3 x

248.

R

cossin24 xx dx;

249.

R

e2xx

2ex

dx;

e2 +1

250.

R

e3x dx

;

ex+2

251.

R

e4x dx

;

ex 1

R

3x

252.

e dx

;

e2x 1

R

x

253.

e

+1

dx;

ex

1

R

x

x

254.

e3 +2e

dx;

e2x+ex+1

255.

R

2x

x

dx;

3ee2x+44e

R

256.R ee5xx+1dx ;

257.R cos2cos4xxdx ;

258.R cosdx4 x ;

259.	R	dx	;
		cosx+2sinx+3

260.R 1+tgsin2xx dx;

261.R ctg5 x dx;

262.R 1+3sindx 2 x ;p

R

263.(3x 1) x2 8x dx;

		x3+x2
264.	R	3	dx;
		x2+6x+10

x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©á⢠...

37

265.

R

5xe2x 3e2xx

dx;

(5x + 3)

;

p

2

+ 3

+ 5

267.

R

e +4 e

dx

aa2xdx+1 (a > 0, a = 1);

266.

x

x

x

R

2x)p

6

dx;

(1

268.

3x2 + 8x

R

px2+5x+17

269.

R

p

6x 10

dx;

270.

2x2 + 4x + 1

dx;

271.

R

p2 x2 dx;

arcsin x

R

3x+1

273.

x

p1 x2

;

272.

R

x2

+10x+1 dx;

R

arcsinx dx

R

274.sin(lnx)dx;

R p

275.3 + 2x x2 dx;

276.R tg3 x dx;

277.R x2 arctg(2x + 1)dx;

R

278.cosmx cosnx dx (m + n =6 0, m n =6 0);

279.	R	ln(cosx)dx					;
		sin2 x
		+p
				x2				9)
280.		ln(xp								dx;
			x 3
281.	R	dx
		2
	R	sin2 3x +1;

282.3+sin5dx x ;

283.R dx ; cos3x+2sin3xR

284.	R	dx				;
		cos2x sin2x+2
285.		dx		;
		2+3cos x2
	R			dx
286.		2sin2 3x			3cos2 3x+1.
	R

x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©á⢠®¯à¥- ¤¥«¥--®£® ¨-â¥£à « ¨ ¥£® ¯à¨«®¦¥-¨ï

2.1. Ž¡é¨¥ ¯®-ïâ¨ï

•ãáâì ç¨á«®¢ ï äã-ªæ¨ï f (x) ®¯à¥¤¥«¥- - [a; b]. Ž¯à¥¤¥«¨¬ à §¡¨¥-¨¥

T ®â१ª [a; b] § ¤ -¨¥¬ ª®-¥ç-®© á¨á⥬ë â®ç¥ª fxig: a = x0 < x1 < : : : < < xi 1 < xi < : : : < xn. „¨ ¬¥â஬ à §¡¨¥-¨ï d(T ) à §¡¨¥-¨ï T ®â१ª [a; b]

- §®¢¥¬ ç¨á«®	d	T		max (x	i	x	i 1	) > 0, 0 < d(T ) 6 b		a. •à¨ § ¤ --®¬
	(		) = i=1;2;:::;n
à §¡¨¥-¨¨ T ®â१ª				[a; b] à áᬮâਬ «î¡ãî á¨á⥬ã â®ç¥ª i 2 [xi 1; xi]
(i = 1; 2; : : : ; n)

(a = x0 6 1 6 x1 < : : : < xi 1 6 i 6 xi < : : : < xn 1 6 n 6 xn = b):

38	x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©á⢠...
• §¡¨¥-¨¥ T ®â१ª	[a; b] ¢¬¥áâ¥ á ¢ë¡à --®© á¨á⥬®© â®ç¥ª =

f 1; 2; : : : ; ng - §®¢¥¬ à §¬¥ç¥--ë¬ à §¡¨¥-¨¥¬ [a; b] ¨ ¡ã¤¥¬ ®¡®§- ç âì T . •® § ¤ --®© äã-ªæ¨¨ f : [z; b] ! R ¨ à §¬¥ç¥--®¬ã à §¡¨¥-¨î T á ¤¨ ¬¥â஬ d(T ) > 0 ¯®áâந¬ á㬬ã

			n
		X
		Sf (T ) =	f ( i)(xi xi 1):
		i=1
Ž¯à¥¤¥«¥-¨¥ 1. ˆ-â¥£à «®¬ •¨¬ -					äã-ªæ¨¨ f (x) -				®â१ª¥ [a; b] - -
§ë¢ ¥âáï â ª®¥ ç¨á«® I (¥á«¨ ®-® áãé¥áâ¢ã¥â), çâ®							lim		Sf (T ) = I.
							d(T )!0		b
	a		b			a
									R
Ž¡®§- ç ¥âáï ¨-â¥£à « •¨¬ -			á«¥¤ãî騬 ®¡à §®¬: I = f (x)dx. ‡¤¥áì
	R		R			R			a
				b			f (x)dx.
¨ ¤ «ìè¥ ¯®« £ ¥¬:		f (x)dx = 0 ¨		f (x)dx =
	a		a			b
‡ ¬¥ç -¨¥. ˆ-â¥£à « •¨¬ -			I =		f (x)dx § ¢¨á¨â®â a, b ¨ f , -® -¥ § -
				a
¢¨á¨â ®â x, ïî饩áï \-¥¬®©" ¯¥à¥¬¥R--®© (¯¥à¥¬¥--®© ¨-⥣à¨à®¢ -¨ï),
x.					R
¢ â® ¢à¥¬ï ª ª -¥®¯à¥¤¥«¥--ë© ¨-â¥£à « f (x)dx = F (x) + C § ¢¨á¨â ®â
•¥ª®â®àë¥ á¢®©áâ¢		¨-â¥£à «	•¨¬ - .
1) •ãáâì äã-ªæ¨¨ f (x) ¨ g(x) ¨-⥣à¨à㥬ë -							[a; b], ⮣¤ ¤«ï «î¡ëå
¨ á¯à ¢¥¤«¨¢ ä®à¬ã« :
b				b			b
Za	( f (x)+ g(x))dx = Za				f (x)dx + Za			g(x)dx:
2) •ãáâì a < c < b, ⮣¤
	b		c			b
	Za	f (x)dx = Za	f (x)dx + Zc			f (x)dx:
«¥--ë© ¨-â¥£à « ®â äã-ªæ¨¨ f (x) -				[a; b], ⮣¤ :			R	f (x)dx | -¥®¯à¥¤¥-
3) •ãáâì f (x) -¥¯à¥àë¢- - [a; b]. •ãáâì F (x) =
	Z	b		b
		f (x)dx = F (x) a = F (b) F (a):
	a
4) •ãáâì äã-ªæ¨ï f (x) -¥¯à¥àë¢- -					[a; b]. •ãáâì äã-ªæ¨¨ '(t) ¨ '0(t)
-¥¯à¥àë¢-ë -	[ ; ], ¯à¨ç¥¬ '( ) = a, '( ) = b ¨ f ('(t)) ®¯à¥¤¥«¥- ¨

x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©á⢠...

39

-¥¯à¥àë¢- -

[ ; ], ⮣¤

b

Za

f (x)dx = Z

f ('(t))'0(t)dt:

5) •ãáâì äã-ªæ¨¨ u(x) ¨ v(x) ¤¨ää¥à¥-æ¨à㥬ë - [a; b],

¨å ¯à®¨§¢®¤-

-ë¥ u0(x) ¨ v0(x) ¨-⥣à¨à㥬ë -

[a; b], ⮣¤

Z

b

b

Z

b

u(x) v0(x)dx = u(x) v(x) a

v(x) u0(x)dx:

a

a

•à¨¬¥à 1. (á¬. ᢮©á⢮ 3).

1

1

= ln(1 + p2) ln1 = ln(1 + p2):

Z

p1 + x2 = ln(x +

) 0

1 + x2

0

dx

p

•à¨¬¥à 2. (á¬. ᢮©á⢮ 3).

2

2

Z

= (cos2 cos0) = 0:

sinx dx = cosx 0

0

•à¨¬¥à 3. (á¬. ᢮©á⢮ 4).

3

Z

p

9 x2 dx;

0

à áᬮâਬ ¯®¤áâ -®¢ªã x = 3sint, t 2 [0; =2]. ˆ¬¥¥¬

3 =2

Z Z

p

p

9 x2 dx = 9 9sin2 t 3cost dt =

00

=2					=2
		9		sin2t			9
= 9 Z	cos2 t dt =		t +		0	=		:
		2		2			4
0

•à¨¬¥à 4. (á¬. ᢮©á⢮ 4).

2 p

Z x2 1 dx; x4

1

40												x2.			Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©á⢠...
																			1
																												x			;			t		0; 3		.
à2	áᬮâਬ ¯®¤áâ=3-®¢ªã1																	cost , â ª ª ª																	2
													x =															=32 [1				2], â®
		p													1																							3			=3			p
					2
				x			1			2			t								sint									2							sin		t
												cos																																	3
Z								dx =	Z	q																dt =			sin		t		cost dt =								0		=					:
				x		4							4							cos				2
										1= cos t															t		Z									3									8
1									0																			0
		•à¨¬¥à 5. (á¬. ᢮©á⢮ 5).
			1										1						1						dx								1
			arctgx dx = x arctgx 0																								= x arctgx 0
Z															Z								x
																							1 + x2
	0															0
																1											1																ln2
																			lnj1 + x2j 0									=				1
																2													4			2 (ln2 ln1) =									4		2		:

2.2.ƒ¥®¬¥âà¨ç¥áª¨¥ ¯à¨«®¦¥-¨ï ®¯à¥¤¥«¥--®£® ¨-â¥£à « I. „«¨- ªà¨¢®©.

I.1. Šà¨¢ ï § ¤ -						ï¢-ë¬ ãà ¢-¥-¨¥¬ y = f (x), x 2 [a; b], ⮣¤ ¤«¨-
ªà¨¢®© ¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:
										l = Z						b
																						1 + [f 0(x)]2 dx:
														a				p
•à¨¬¥à 6. • ©â¨ ¤«¨-㠪ਢ®© y =																															x2			-					ãç á⪥ x 2 [0; 4].
																															6
																									x2		0								x
													y0					=									=
																									6									3
					4																						4
						r1 + 3																																	32 + x2 dx =
		l = Z																			dx = 3 Z
															x					2							1
			0																													0			p
																																												4
= 3					2 x																																									) 0			=
										32 + x2 + 2 ln(x +																													32 + x2
1					1		p																	9												p
			1			1																					9
														2								2																				2				2
=							4		3							+ 4								+						ln(4 +									3					+ 4 )
																											2
			3 2						p																														p
2 0 p32					+ 0 2 ln(0+ p32 + 0) = 3																																							10+ 2 ln9
	1								9																														1										9
															10 + 2 ln3 =
		2			ln3 = 3																																		3 +					2 ln3:
9									1															9															10					3