заочникам / мат анализ / интегр исчисление

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

21

1.6.ˆ-⥣à¨à®¢ -¨¥ ¢ëà ¦¥-¨©, ᮤ¥à¦ é¨å âਣ®-®¬¥âà¨ç¥- ᪨¥ äã-ªæ¨¨

I.ˆ-⥣à¨à®¢ -¨¥ ¢ëà ¦¥-¨© ¢¨¤

Z

R(sinx; cosx)dx;

£¤¥ R | à 樮- «ì- ï äã-ªæ¨ï.

‚ í⮬ á«ãç ¥ áãé¥áâ¢ã¥â ã-¨¢¥àá «ì-ë© ¯à¨¥¬ à 樮- «¨§ 樨 ¢ëà - ¦¥-¨©, áâ®ïé¨å ¯®¤ §- ª®¬ ¨-â¥£à « . € ¨¬¥--®, ¨á¯®«ì§ãï ¯®¤áâ -®¢ªã

t = tg x2 ( < x < ) ¨ âਣ®-®¬¥âà¨ç¥áª¨¥ ä®à¬ã«ë:

2tg x

2t

sinx =

2

=

;

1 + tg2 x

1 + t2

2

cosx =

1 tg2 x2

=

1 t2

;

1 + tg2 x

1 + t2

2

x = 2arctgt;

dx =

2dt

;

1 + t2

¯®«ã稬 ¯®¤ ¨-â¥£à «®¬ à 樮- «ì-ãî äã-ªæ¨î.

•à¨¬¥à 32.

dx

1

d3x

Z

=

Z

=

cos3x + 2sin3x

3

cos3x + 2sin3x

1

dz

1

2dt

=

Z

=

Z

=

3

cosz + 2sinz

3

(1 + t2)

1+1 tt22

+

4t

1+t2

2

dt

2

dt

2

du

=

Z

=

Z

=

Z

=

3

1 t2 + 4t

3

5 (t 2)2

3

5 u2

p

p

+ C =

tg

3x

2 +

2

1

5 + u

1

2

5

=

2p

ln

p

3p

ln

p

+ C:

3

tg

3x

5

5

u

5

2

5

2

•®¤áâ -®¢ª , 㯮¬ï-ãâ ï ¢ëè¥, ¯à¨¢®¤¨â

ç áâ® ª

á«®¦-ë¬ ¯à¥®¡à

§®-

¢ -¨ï¬, ¯®í⮬㠢 -¨¦¥¯à¨¢¥¤¥--ëå ç áâ-ëå á«ãç ïå ¯à¥¤¯®çâ¨â¥«ì-ë á«¥¤ãî騥, ¡®«¥¥ 㤮¡-ë¥ ¯à¨¥¬ë:

) ¯ãáâì R(u; v) = R( u; v), ⮣¤ à 樮- «¨§ æ¨ï ¤®á⨣ ¥âáï ¯®¤áâ - -®¢ª®© t = cosx;

¡) ¯ãáâì R(u; v) = R(u; v), ⮣¤ à 樮- «¨§ æ¨ï ¤®á⨣ ¥âáï ¯®¤áâ - -®¢ª®© t = sinx;

¢) ¯ãáâì R(u; v) = R( u; v), ⮣¤ à 樮- «¨§ æ¨ï ¤®á⨣ ¥âáï ¯®¤áâ - -®¢ª®© t = tgx.

22	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

•à¨¬¥à 33.

Zsinx

dx;

2

4 + cos x

R(sinx; cosx) =

sinx

;

4 + cos2 x

R( sinx; cosx) =

sinx

;

4 + cos2 x

R(sinx; cosx) = R( sinx; cosx);

¯®í⮬㠨ᯮ«ì§ã¥¬ ¯®¤áâ -®¢ªã t = cosx:

Z

sinx

dx = Z

d cosx

= Z

dt

1

arctg

cosx

+ C:

=

4 + cos2 x

4 + cos2 x

4 + t2

2

2

•à¨¬¥à 34.

Z

dx

;

sin2 x 5sinx cosx

R(sinx; cosx) =

1

;

sin2 x 5sinx cosx

R( sinx; cosx) =

1

;

sin2 x 5sinx cosx

R(sinx; cosx) = R( sinx; cosx);

¨á¯®«ì§ã¥¬ ¯®¤áâ -®¢ªã t = tgx:

dx

Z

cos2 x

dx

cos2 x

Z sin x 5sinx cosx

cos2 x

=

=

2

d tgx

dt

5sinx cosx

dt

sin2 x

= Z

= Z

= Z

=

tg2 x 5tgx

t2 5t

t 25 2

25

2

1

25 + t

25

1

t

5

25

t +

25

5

t

5

=

ln

=

ln

1

5

1

=

ln 1

+ C

=

ln

1

5ctgx

+ C:

5

t

j

5

j

II. ˆ-â¥£à «ë ¢¨¤

sin

m

x cos

n

x dx.

•à¨ ¨-⥣à¨à®¢ -¨¨R íâ¨å ¢ëà ¦¥-¨© ®ª §ë¢ îâáï ¯®«¥§-묨 á«¥¤ãî-

騥 âਣ®-®¬¥âà¨ç¥áª¨¥ ä®à¬ã«ë

sinx cosx =

sin2x

;

sin2 x =

1 cos2x

;

cos2 x =

1 + cos2x

:

2

2

2

1. • áᬮâਬ ¨-â¥£à «ë ¢¨¤ :

R sinm x dx, R cosm x dx, ¯®« £ ¥¬, çâ®

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

23

) m > 0, ç¥â-®¥, m = 2k.

Z	sinm x dx = Z	(sin2 x)k dx = Z	(1	cos2x)k	dx =	1	Z (1 cos2x)k d2x:
				2k		2k+1

¡) m > 0, -¥ç¥â-®¥, m = 2k + 1.

Z	sinm x dx =	Z	sin2k x sinx dx =
	= Z	(sin2 x)k d cosx = Z (1 cos2 x)k d cosx = Z (1 t2)k dt;

£¤¥ t = cosx.

‚®§¢¥¤ï ¢ëà ¦¥-¨ï, áâ®ï騥 ¯®¤ ¨-â¥£à « ¬¨ ¯® ¡¨-®¬ã •ìîâ®-

¢ á®-

®в¢¥вбв¢гойго бв¥¯¥-м, ¯®«гз ¥¬ ап¤ ¨-в¥£а «®¢ в¨¯

( ) ¨ (¡), -® á -¨§-

訬¨ á⥯¥-ﬨ, ¨å ¤ «ìè¥ ã¯à®é ¥¬ ⥬ ¦¥ ®¡à §®¬.

‘«ãç ©

cosm x dx ¤¥« ¥âáï

- «®£¨ç-®.

•à¨¬¥àR

35.

Z

cos4 x dx = Z (cos2 x)2 dx = Z

(1+ cos2x)2

d2x =

8

1

Z (1 + 2cosz + cos2 z)dz =

z

sinz

=

+

+

8

8

4

1

Z

1 + cos2z

3

sin2x

sin4x

+

dz =

x +

+

+ C:

8

2

8

4

32

•à¨¬¥à 36.

Z

cos3 x dx = Z

cos2 x d sinx = Z (1 sin2 x)d sinx =

sin3 x

= Z (1 t2)dt = t

t3

+ C = sinx

+ C:

3

3

¢) m = 1

dx

dx

Z

Z

;

:

sinx

cosx

Z	dx	= Z	dx
	sinx		2sin x2 cos x2

Z	cos2	dx	cos x
		2
=		2
		x	sin x2
			2

= Z	d tg x	= ln tg	x			+ C;
	tg x2		2
	2

Z

dx

Z

d x +

x

cosx

sin x + 2

2

4

=

2

= ln

tg

+

+ C:

24	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

£) m < 0, jmj = 2k.

dx

1

k 1

dx

Z

cosm x dx = Z

=

Z

=

cos2k x

cos2 x

cos2 x

2 x + cos2 x

k 1

= Z

sin

d tgx =

Z (1 + tg2 x)k 1 d tgx =

cos2 x

- «®£¨ç-®

= Z

(1+ t2)k 1 dt; t = tgx;

Z

sinm x dx = Z

dx

= Z

(1+ ctg2 x)k 1 d ctgx:

sin2k x

•à¨¬¥à 37.
Z	dx		= Z	1		dx	=
	cos4 x			cos2 x		cos2 x

=

¤) m < 0, jmj = 2k + 1.

Z sin	2 x + cos2 x	d tgx =
	cos2 x

Z (t2 + 1)dt = t3 + t + C 3

Z

(tg2 x + 1)d tgx =

= tg3 x + tgx + C: 3

Z

cosm x dx = Z

dx

Z

cosx

=

dx =

cos2k+1 x

cos2k+2 x

= Z

d sinx

=

Z

dt

(1 sin2 x)k+1

(1 t2)k+1

¤ «•à¨¬¥¥ ¥à 38.

R

dx

¤¥« ¥âáï - «®£¨ç-®.

sin

2k+1

x

á¬. ¯ã-ªâ 1.4. ‘«ãç ©

dx

cosx dx

d sinx

d sinx

dt

Z

= Z

=

Z

= Z

=

Z

;

cos3 x

cos4 x

(cos2 x)2

(1 sin2 x)2

(1 t2)2

(¤ «¥¥ á¬. ¯ã-ªâ 1.4).

‚á«ãç ¥ ¤) ¤«ï ¯®-¨¦¥-¨ï á⥯¥-¨ 㤮¡-® ¯à¨¬¥-ïâì ä®à¬ã«ã sin2 x +

+cos2 x = 1.

Z

dx

= Z

sin

2 x + cos2 x

dx =

cos3 x

cos3 x

=

sin2 x

1

dx = ln tg

x

+

Z

sinx

Z cos3 x

+ cosx

2

4

cos3 x d cosx:

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

25

ˆ-⥣à¨àãï ¯®á«¥¤-¨© ¨-â¥£à « ¯® ç áâï¬, ¯®«ã稬:

Z

sinx

1

Z

sinx d cos 2 x =

d cosx =

cos3 x

2

cos2 x Z

cosx =

= 2

cos2 x Z

cos2 x

=

2

1

sinx

d sinx

1

sinx

dx

1 sinx

1

tg

x

:

=

ln

+

2 cos2 x

2

2

4

Žª®-ç ⥫ì-® ¯®«ã稬:

dx

=

1

ln tg

x

+

+

1 sinx

Z cos3 x

2

2

4

2 cos2 x:

2) • áᬮâਬ ¨-â¥£à «ë ¢¨¤

Z

sinm x cosn x dx;

m > 0; n > 0:

¥) ®¤¨- ¨§ ¯®ª § ⥫¥© m ¨«¨ n -¥ç¥â-ë©, ¡¥§ ¨§¬¥-¥-¨ï ®¡é-®á⨠¯®-

« £ ¥¬ n = 2k + 1.

sinm x cos2k+1 x dx = Z

Z

sinm cosn x dx = Z

sinm x(cos2 x)k d sinx =

= Z

sinm x(1 sin2 x)k d sinx = Z

tm(1 t2)k dt;

£¤¥ t = sinx ¨ ¤ «¥¥ ¯® ¡¨-®¬ã •ìîâ®- .

•à¨¬¥à 39.

Z

sin2 x cos3 x dx = Z

sin2 x cos2 x d sinx =

= Z

sin2 x(1 sin2 x)d sinx = Z

t2(1 t2)dt =

= Z (t2 t4)dt =

t3

t5

+ C =

sin3 x

sin5 x

+ C:

3

5

3

5

¦) m ¨ n ç¥â-ë¥, â® ¥áâì m = 2k, n = 2l,

Z

sinm x cosn x dx = Z

sin2k x cos2l x dx = Z

sin2k x(1 sin2 x)l dx;

¤ «¥¥, ¨á¯®«ì§ãï ¡¨-®¬ •ìîâ®- , ᢮¤¨¬ ª á«ãç î ).

•à¨¬¥à 40.

sin2(1 sin2 x)dx = Z

sin2 x dx Z

Z

sin2 x cos2 x dx = Z

sin4 x dx:

26

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

3. ˆ-â¥£à «ë ¢¨¤

m

m

sinn x dx ¨

cosn x dx.

cos x

§) m = n, ¨á¯®«ì§ãïRä®à¬ã«ã tgR2 xsin=x

1

1.

cos2 x

sinm x

1

Z

dx = Z

tgm 2

1 dx =

cosm x

cos2 x

tgm 1 x

= Z

tgm 2 d tgx Z

tgm 2 x dx =

Z

tgm 2 x dx;

m 1

¯®«ã稬 ä®à¬ã«ã ¯®-¨¦¥-¨ï. €- «®£¨ç-®

cosm x

1

Z

dx = Z

ctgm x dx =

ctgm 1 x Z

ctgm 2 x dx:

sinm x

m 1

•à¨¬¥à 41.

sin3 x

tgx

1

1 dx =

Z

dx = Z

tg3 x dx = Z

cos3 x

cos2 x

2 x

sinx

= Z

tgx d tgx Z

tgx dx =

tg

Z

dx =

2

cosx

tg2 x

d cosx

tg2 x

=

+ Z

=

+ lnj cosxj + C:

x

cosx

2

¨) ¯®ª § ⥫ì ç¨á«¨â¥«ï m | ç¥â-ë©, m = 2k.

Z	cosm x	dx =	Z	cos2k x	dx =	Z	(1 sin2 x)k	dx:
	sinn x			sinn x			sinn x

ˆ-â¥£à « à ᯠ¤ ¥âáï - á㬬㠨-â¥£à «®¢ ¢¨¤ (1).

•à¨¬¥à 42.

Z

cos4 x

dx =

Z

(1 sin2 x)2

dx =

sin3 x

sin3 x

= Z

1

2sin2 x + sin4 x

dx = Z

dx

2Z

dx

+ Z

sinx dx:

sin3 x

sin3 x

sinx

ª) ¯®ª § ⥫ì ç¨á«¨â¥«ï m -¥ç¥â-ë©, m = 2k + 1.

Z	cosm x	dx =	Z	cos2k+1 x	dx =	Z	(1 sin2 x)k		d sinx:
	sinn x			sinn x			sinn x
ˆ-â¥£à « à ᯠ¤ ¥âáï -			®â¤¥«ì-ë¥ á« £ ¥¬ë¥ ¢¨¤					R tm dt.

x1.

•¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

27

•à¨¬¥à 43.

Z

cos3 x

dx =

Z

(1 sin2 x)

d sinx =

dt

dt =

sin2 x

sin2 x

Z t2 Z

1

1

=

t + C =

sinx + C;

t

sinx

£¤¥ t = sinx.

R

dx

4. ˆ-â¥£à «ë ¢¨¤

, m > 0, n > 0.

cosm x sinn x

Z

dx

= Z

cos2 x + sin2 x

dx =

cosm x sinn x

cosm x sinn x

dx

dx

= Z

+ Z

;

cosm 2 x sinn x

cosm x sinn 2 x

¯®¢â®à¨¢ íâ®â ¯à¨¥¬ âॡ㥬®¥ ª®«¨ç¥á⢮ à §, ᢮¤¨¬ ¨-â¥£à « ª ¯à¥¤ë- ¤ã騬 ⨯ ¬.

•à¨¬¥à 44.

Z

dx

= Z

cos2 x + sin2 x

dx =

cos3 x sin2 x

cos3 x sin2 x

dx

dx

= Z

+ Z

=

cosx sin2 x

cos3 x

+ Z

dx

= Z

cos3 x

Z		cos2 x + sin2 x
						dx+
		cosx sin2 x
	cosx				dx				dx
			dx + Z					+ Z		:
	sin2 x			cos3 x					cos3 x

…᫨ m + n = 2k, â® ¯à®é¥ ¤¥«¨âì ç«¥-ë ¤à®¡¨ -

cosm+n x.

Z

dx

Z

dx

cosm x sin x

cosm+n x

cosm+x x

n

=

cosm x sinn x

2

=

dx

k

1

n

= Z

= Z (1 + tg

x)

tg

x d tgx =

cos2k x tgn x

= Z

t n(1 + t2)k 1 dt;

£¤¥ t = tgx.

28	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

•à¨¬¥à 45.

Z

dx

= Z

dx

=

cosx sin3 x

cos4 x tg3 x

= Z

(1+ tg2 x)tg 3 x d tgx = Z

t 3(1+ t)dt =

= Z (t 3 + t 2)dt =

1

1

+ C:

2tg2 x

tgx

III. ˆ-â¥£à «ë ¢¨¤

Z

sin(ax + b)cos(cx + p)dx;

Z

sin(ax + b)sin(cx + p)dx;

Z

cos(ax + b)cos(cx + p)dx

ã¯à®é îâáï - ®á-®¢ -¨¨ âਣ®-®¬¥âà¨ç¥áª¨å ⮦¤¥á⢠sin cos = 12 (sin( + ) + sin( )); sin sin = 12 (cos( ) cos( + )); cos cos = 12 (cos( + ) + cos( )):

•à¨¬¥à 46.

Z sin(3x + 1)cos(2x + 3)dx = 12 Z (sin(5x + 4)+

+ sin(x 2))dx = 1 Z sin(5x + 4) d(5x + 4)+

25

	sin(x 2)d(x 2)		x			cos(	x
+ Z		=	cos(5 + 4)				2)	+ C:
			10				2

Šà®¬¥ ⮣®, ¯à¨ ¨-⥣à¨à®¢ -¨¨ âਣ®-®¬¥âà¨ç¥áª¨å äã-ªæ¨© ¬®¦-®

¯®«ì§®¢ âìáï ä®à¬ã« ¬¨ •©«¥à
sinx =	1	(eix e ix);	cosx =	1		(eix + e ix):
	2i			2

‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï				29
IV. ˆ-â¥£à «ë ¢¨¤			Z
Z	P (x)sinax dx;			P (x)cosax dx;
	Z	ebx sinax dx;	Z	ebx cosax dx;
	Z	ebx sinax dx;	Z	ebx cosax dx;

£¤¥ P (x) | ж¥«л© ¬-®£®з«¥-, ¨-в¥£а¨аговбп ¯® з бвп¬, б¬. ¯г-ªв 1.3.

‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï

•©â¨ ¨-â¥£à «ë:

1.R (x2 + 3x3 + x + 1)dx;

2.R x4 + p5 x + 3px + x12 dx R x1 5 dx;

4.

R x4

dx;

dx;

px

px3

3.

1

1

4

R

5x8+1

px4

5.

R

x5 1

dx;

6.

x1

+

1

+

1

dx;

x2

x3

7.

R

x

dx;

R

(p

x

1)3

8.

(2x + 3x)dx;

dx;

9.

R

4x

3 + p4 x3

R

x

10.	R ex 2	e x	dx;
		x3

11.R 1+2x2 p13 x2 dx;

12.R 1+x4x2 dx;

13.	R ex 1 +	e x	dx;
		cos2 x
	R

14.(sinx + 5cosx)dx;

	R	cos2x
16.		sin2 x cos2 x ;
15.		2 x sin2 x dx;
		cos		dx
		3tg	2
	R			x+4
18.		3 cos2 x						dx;
17.	R	sin2 x					dx;
	R	2ctg2 x
19.		sin		x
	R

20.tg2 x dx;

21.R ctg2 x dx;

22.R sin2 x2 dx;R

30										‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï
24.	R	sin x2 cos x2								2 dx;
23.		cos2		x dx;
	R			2
				2 p				2
25.	R	2xex dx;
		p
26.			1+px		1 x41 x				dx;
	R

R

27.cos5x dx;

28.	R	sin7x dx;
29.		cos x4 dx;
30.	R	e x dx;
	R				x					x
	R
31.			e 2 + e							2 dx;
32.	R			dx			;
			2
			cos 3x
33.			dx			;
			sin2 x3
34.	R	(2 + 5x)9 dx;
35.	R		p	dx				;
				2 3x
	R	p
	R

36.R p2x 5dx;

37.3 3 7x dx;

38.R 5xdx+2;

39.R 2dx3x ;

40.R xx2dx+3;

R

41.ctgx dx;

42.	R	tgx dx;
43.	R		sinx dx		;
			1+3cosx
44.			dx			;
			x(1+lnx)5
45.	R		cos3x dx	;
			3+sin3x
46.	R		cos2x
			sinx cosx dx;
	R

47.sin2 x cosx dx;

48.R cos3 x sinx dx;

49.R ecosx sinx dx;

50.R e x3x2 dx;R

p

x

51.

R

e p

dx

;

x

52.

cosx dx

;

sin3 x

53.

R

sinx dx

;

cos5 x

R

3

54.

p2+lnx

dx;

x

55.

R

p3 + cos5x sin5x dx;

R

56.

R

cos3x dx

;

7

p3+5sin3x

Re4x

57.5+2e4x dx;