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ˆ-â¥£à «ì-®¥ ¨áç¨á«¥-¨¥ äã-ªæ¨¨ ®¤-®© ¯¥à¥¬¥--®©

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «. Žá-®¢-ë¥ á¢®©á⢠-¥- ®¯à¥¤¥«¥--®£® ¨-â¥£à «

1.1. Ž¡é¨¥ ¯®-ïâ¨ï

Ž¯à¥¤¥«¥-¨¥ 1. ”ã-ªæ¨ï F (x) - §ë¢ ¥âáï ¯¥à¢®®¡à §-®© ¤«ï äã-ªæ¨¨ f (x) - (a; b), ¥á«¨ ¤«ï «î¡®£® x 2 (a; b) F 0(x) = f (x).

•à¨¬¥à 1. ”ã-ªæ¨ï sin(5x 1) ¥áâì ¯¥à¢®®¡à §- ï ¤«ï äã-ªæ¨¨ 5cos(5x 1) - ¢á¥© ç¨á«®¢®© ¯àאַ©, â ª ª ª (sin(5x 1))0 = 5cos(5x 1).

…᫨ äã-ªæ¨ï f (x) ¨¬¥¥â - (a; b) ¯¥à¢®®¡à §-ãî F0(x), â® ¬-®¦¥á⢮ ¢á¥å ¯¥à¢®®¡à §-ëå äã-ªæ¨¨ f (x) - (a; b) ᮢ¯ ¤ ¥â á ¬-®¦¥á⢮¬ äã-ªæ¨© F (x) = F0(x)+ C, £¤¥ C | «î¡ ï ¯®áâ®ï-- ï.

Ž¯à¥¤¥«¥-¨¥ 2. •¥®¯à¥¤¥«¥--ë¬ ¨-â¥£à «®¬ ®â äã-ªæ¨¨ f (x) - (a; b) - §ë¢ ¥âáï ¬-®¦¥á⢮ ¢á¥å ¯¥à¢®®¡à §-ëå F (x) (¥á«¨ ®-¨ áãé¥áâ¢ãîâ)

äã-ªæ¨¨ f (x) -

(a; b). •¥®¯à¥¤¥«¥--ë© ¨-â¥£à « ®â f (x) -

(a; b) ®¡®§- -

ç ¥âáï ᨬ¢®«®¬

R

3

f (x)dx; f (x) - §ë¢ ¥âáï ¯®¤ë-â¥£à «ì-®© äã-ªæ¨¥©.

2

x3

•à¨¬¥à 2. •ãáâì f (x) = x

. ”ã-ªæ¨ï F (x) =

¥áâì ¯¥à¢®®¡à §- ï ¤«ï

äã-ªæ¨¨ f (x) = x2 -

¯à®¬¥¦ã⪥ (

;+

), â ª ª ª

x3

0

= 3x2 = x2.

1

1

3

•®í⮬ã

3

Z

x2 dx =

x3

+ C:

3

•à¨¬¥à 3. •ãáâì f (x) =

1. •¥à¢®®¡à §-®© f (x) =

1

-

¯à®¬¥¦ã⪥

x

x

(0; +1) ï¥âáï äã-ªæ¨ï F (x) = lnx,

-

¯à®¬¥¦ã⪥ ( 1;0) äã-ªæ¨ï

F (x) = ln( x). ’ ª¨¬1

®¡à §®¬, äã-ªæ¨ï F (x) = lnjxj ¥áâì ¯¥à¢®®¡à §- ï

¤«ï äã-ªæ¨¨ f (x) = x

- «î¡®¬ ¯à®¬¥¦ã⪥, -¥ ᮤ¥à¦ 饬 0. •®í⮬ã

Z

dx

= lnjxj + C:

x

Žá-®¢-ë¥ á¢®©áâ¢

-¥®¯à¥¤¥«¥--®£® ¨-â¥£à « :

R

1)d f (x)dx = f (x)dx;

2)R f (x)dx 0 = f (x);

3)R df (x) = R f 0(x)dx.

1

2		x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...
Žá-®¢-ë¥ ¯à ¢¨« ¢ëç¨á«¥-¨ï -¥®¯à¥¤¥«¥--ëå ¨-â¥£à «®¢.
•ãáâì -	(a; b) áãé¥áâ¢ãîâ -¥®¯à¥¤¥«¥--ë¥ ¨-â¥£à «ë			g(x)dx ¨
f1)(x)dx. ’®£¤	¤«ï «î¡ëå ¨ - (a; b) á¯à ¢¥¤«¨¢		ä®à¬ã«	R:
R
Z ( f (x)+ g(x))dx = Z		f (x)dx + Z	g(x)dx:

2) •ãáâì äã-ªæ¨¨ u(x), v(x) ¨¬¥îâ -¥¯à¥àë¢-ë¥ ¯à®¨§¢®¤-ë¥ u0(x), v0(x) - ¯à®¬¥¦ã⪥ (a; b). ’®£¤ á¯à ¢¥¤«¨¢® à ¢¥-á⢮:

¨«¨

Z

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

v(x)u0(x)dx

Z

u dv = uv Z

v du:

R

=

( )+ . •ãáâì äã-ªæ¨ï

= ( ) ¨¬¥¥â -¥¯à¥àë¢-ãî

0

3)

•ãáâì -

( ; ) áãé¥áâ¢ã¥â

-¥®¯à¥¤¥«¥--ë© ¨-â¥£à «

f (t)dt =

F t

C

t

u x

¯à®¨§¢®¤-ãî u (x)

-

(a; b) ¨ u((a; b)) ( ; ). ’®£¤ -

(a; b) á¯à ¢¥¤«¨¢® à ¢¥-á⢮:

Z

f (u(x))u0(x)dx = Z

f (u(x))du(x) = F (u(x))+ C = Z

f (t)dt :

4) •ãáâì áâண® ¬®-®â®-- ï äã-ªæ¨ï x = !(t) ¨¬¥¥â -¥¯à¥àë¢-ãî ¯à®¨§-

¢®¤-ãî !0(t) -

( ; ) ¨ !(t) : ( ; ) ! (a; b). •ãáâì äã-ªæ¨ï f (x) ®¯à¥¤¥«¥-

-

(a; b) ¨ áãé¥áâ¢ã¥â -¥®¯à¥¤¥«¥--ë© ¨-â¥£à «

Z

f (!(t))!0(t)dt = Z

g(t)dt = G(t) + C:

’®£¤ - (a; b) ¨¬¥¥¬:

Z Z Z

f (x)dx = f (!(t))d!(t) = f (!(t))!0(t)dt = G(t) + C = G(v(x))+ C;

£¤¥ t = v(x) | äã-ªæ¨ï, ®¡à â- ï ª äã-ªæ¨¨ x = !(t).

	’ ¡«¨æ ®á-®¢-ëå -¥®¯à¥¤¥«¥--ëå ¨-â¥£à «®¢.
1)	R	0 dx = C;
2)	R	1dx = x + C;
3)	R x dx = x +1+1 + C ( 6= 1);
	R

x= lnjxj + C (x =6 0);

5)R ax dx = lnaxa + C (0 < a =6 1), R ex dx = ex + C;

R

6)sinx dx = cosx + C;4) dx

R

7)cosx dx = sinx + C;

8)R cosdx2 x = tgx + C (x =6 2 + n; n = 0; 1; 2;: : :);

9)R sindx2 x = ctgx + C (x =6 n; n = 0; 1; 2;: : :);

x1.

•¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

3

10)

R

dx

= a1

arctg xa + C

(a 6= 0);

a2+x2

11)

dx

=

1

ln aa+xx

+ C

(a 6= 0; jxj 6= jaj);

a2 x2

2a

dx

x

12)

R

p

= arcsin a

+ C

(a = 0;

x

< a

);

a2

x2

R

dx

6

j j

j j

13)

px2+k

= ln

x + px2 + k

+ C

(k = 0, ¢ á«ãç ¥ k < 0

x

> k

);

R

j

j

6

j j

j j

R

14)shx dx = chx + C;

R

15)

chx dx = shx + C;

16)

R

dx

= thx + C;

2

17)

ch x

=

cthx + C; (x = 0).

shdx2 x

’R

6

¡«¨æ1

®á-®¢-ëå ¤¨ää¥à¥-æ¨ «®¢.

ax

b

) (

a

6= 0);

1) dx = a d( p+1+

2) xp dx =

dx

(p 6= 1);

p+1

3) dxx

= d(lnjxj)

(x 6= 0);

4) sinx dx = d cosx;

5) cosx dx = d sinx;

6)

dx

= d tgx;

2

x

cos

7)

dx

= dxctgx;

sin2 x

8) ax dx =

da

;

ex dx = dex;

lna

9)shx dx = d chx;

10)chx dx = d shx;

11)p1dxx2 = d arcsinx = d arccosx;

12)1+dxx2 = d arctgx = d arcctgx.

1.2. ˆ-⥣à¨à®¢ -¨¥ ¯ã⥬ § ¬¥-ë ¯¥à¥¬¥--ëå

Ž¤¨- ¨§ - ¨¡®«¥¥ à á¯à®áâà -¥--ëå ¬¥â®¤®¢, ¯à¨¬¥-塞ëå ¯à¨ ¢ëç¨- á«¥-¨¨ -¥®¯à¥¤¥«¥--ëå ¨-â¥£à «®¢ | ¬¥â®¤ § ¬¥-ë ¯¥à¥¬¥--ëå. ‚ ¥£® ®á-®¢¥ «¥¦ â ¯à ¢¨« 3 ¨ 4, áä®à¬ã«¨à®¢ --ë¥ ¢ ¯à¥¤ë¤ã饬 ¯ã-ªâ¥. Œ¥- ⮤ ¯®¤áâ -®¢ª¨ á®á⮨⠢ ⮬, çâ® á®®¡à §-® ¢¨¤ã ¯®¤ë-â¥£à «ì-®© äã-ª- 樨 á®áâ ¢«ïî⠢ᯮ¬®£ ⥫ì-ãî äã-ªæ¨î, ¯®¤áâ -®¢ª ª®â®à®© ¢ ¨á室- -ë© ¨-â¥£à « ¯à¨¢®¤¨â ¥£® ª ¢¨¤ã, ¡®«¥¥ 㤮¡-®¬ã ¤«ï ¨-⥣à¨à®¢ -¨ï.

‚л¤¥«повбп ¤¢¥ д®а¬л ¯®¤бв -®¢ª¨.
I. •ãáâì âॡã¥âáï ¢ëç¨á«¨âì ¨-â¥£à «			u	g(x)dx. ‘®£« á-® ¯à ¢¨«ã 3
¢ë¡¥à¥¬, ¥á«¨ í⮠㤠¥âáï, â ªãî äã-ªæ¨î			R(		), çâ® ¯®¤ë-â¥£à «ì-®¥ ¢ë-
à ¦¥-¨¥ ¯à¥¤áâ ¢«ï¥âáï ¢ ¢¨¤¥:
Z	g(x)dx = Z	f (u(x))u0(x)dx = Z			f (u(x))du(x):

’®£¤ , ¤¥« ï § ¬¥-ã ¯¥à¥¬¥--ëå t = u(x), ¯® ᪠§ --®¬ã ¢ëè¥, ¤®áâ â®ç-®

4		x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...
- ©â¨ ¨-â¥£à «
Z	f (t)dt;	t = u(x):

ˆ§«®¦¥--ë© ¬¥â®¤ ¯à¨¬¥-ï¥âáï ¯à¨ ¢ëç¨á«¥-¨¨ ¨-â¥£à «®¢ ¢¨¤ :

R

1)f (ax + b)dx,

¤¥« ï § ¬¥-ã t = ax + b, ¯®«ã稬:

Z f (ax + b)dx = a1 Z f (ax + b)d(ax + b) = a1 Z f (t)dt;

R

2)f (sinx)cosx dx,

¤¥« ï § ¬¥-ã t = sinx, ¯®«ã稬:

Z Z Z

f (sinx)cosx dx = f (sinx)d sinx = f (t)dt;

R

3)f (cosx)sinx dx,

¤¥« ï § ¬¥-ã t = cosx, ¯®«ã稬:

Z Z Z

f (cosx)sinx dx = f (cosx)d cosx = f (t)dt;

¤¥« ïR§ ¬¥-ã t = tgx, ¯®«ã稬:
4) f (tgx)	1		dx,
	2	x
	cos
	Z	f (tgx)		dx	=
				cos2 x

(lnx)1 dx,
¤¥«5)ïR§ ¬¥-ãx t = lnx, ¯®«ã稬:
Z	f (lnx)	dx		= Z
		x

6) R f (ex)ex dx,

¤¥« ï § ¬¥-ã t = ex, ¯®«ã稬:

Z

Z Z

f (tgx)d tgx = f (t)dt;

Z

f (ln(x))d lnx = f (t)dt;

Z Z

f (ex)ex dx =

f (ex)dex =

f (t)dt;

¤¥«7)ïR§(

ax2 + b)px dx,

¬¥-ã t = ax2 + b, ¯®«ã稬:

Z

1

Z (ax2 +b)p dx2

1

Z (ax2

1

Z

(ax2

+b)px dx =

=

+b)p d(ax2 +b) =

tp dt:

2

2a

2a

Z cosx

sinx cos2 x dx =

x1.

•¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

5

•à¨¬¥à 4.

dx

1

d(5x + 2)

Z

p

=

Z

p

=

5

5x + 2

5x + 2

t 1=2 dt = 5 t1=2

+ C =

5 p5x + 2 + C:

= 5 Z

pt

= 5 Z

1

dt

1

2

2

•à¨¬¥à 5.

Z sin(3x 4)dx = 13 Z sin(3x 4)d(3x 4) =

= 13 Z sint dt = 13 cost + C = 13 cos(3x 4)+ C:

•à¨¬¥à 6.

dx

=

1

d(2x 6)

=

1 dt

=

1

ln t

+ C =

1

ln

2x

6

+ C:

Z

2x 6

2

Z

2x 6

2 Z t

2

2

j

j j

j

•à¨¬¥à 7.

Z

sin3 x cosx dx = Z

sin3 x d sinx = Z

t3 dt =

t4

+ C =

sin4 x

+ C:

4

4

•à¨¬¥à 8.

Zsinx

4 cosx dx =

•à¨¬¥à 9.

Zdx

sinx cosx =

•à¨¬¥à 10.

Zd( cosx) = 4 cosx

Z

=

Z	d( cosx + 4)				=
		4 cosx
= Z			dt
				= lnjtj + C = lnj4 cosxj + C:
			t

Z	1			dx			=
	tgx		cos2 x
d tgx	= Z			dt		= lnjtj + C = lnj tgxj + C:
tgx				t

Z	lnx		dx = Z	lnx d lnx = Z		t2		1
					t dt =		+ C =		ln2 x + C:
	x					2		2

6

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

•à¨¬¥à 11.

e2x

1

e2x

1

de2x

1

d(e2x + 1)

Z

dx =

Z

d2x =

Z

=

Z

=

e2x + 1

2

e2x + 1

2

e2x + 1

2

e2x + 1

1

Z

dt

1

1

=

=

lnjtj + C =

ln(e2x + 1)+ C:

2

t

2

2

•à¨¬¥à 12.

x dx

=

1

dx2

=

1

d(5x2 3)

=

Z

5x2 3

2 Z

5x2 3

2 5 Z

5x2 3

1

Z

dt

1

1

lnj5x2 3j + C:

=

=

lnjtj + C =

10

t

10

10

II. •а¨¬¥-пп ¢в®аго д®а¬г ¯®¤бв -®¢ª¨, ¯®«м§говбп ¯а ¢¨«®¬ 4. ‚ ¯®- ¤л-в¥£а «м-®¥ ¢ла ¦¥-¨¥ -¥¯®ба¥¤бв¢¥--® ¯®¤бв ¢«пов¢¬¥бв® x дг-ªж¨о

x = !(t),	¨¬¥--®:	f (!(t))d!(t) = Z	f (!(t))!0(t)dt = Z
Z	f (x)dx = Z			g(t)dt;

£¤¥ g(t) | ¡®«¥¥ 㤮¡- ï ¤«ï ¨-⥣à¨à®¢ -¨ï äã-ªæ¨ï, 祬 f (x). •à¨ í⮬ - äã-ªæ¨î !(t) - ª« ¤ë¢ îâáï ãá«®¢¨ï áâண®© ¬®-®â®--®áâ¨, çâ® ®¡¥á- ¯¥ç¨¢ ¥â áãé¥á⢮¢ -¨¥ ®¡à â-®© äã-ªæ¨¨ t = v(x) ¨ ¯à¥¤áâ ¢«¥-¨¥:

Z Z

f (x)dx = g(t)dt = G(t) + C = G(v(x))+ C:

•à¨¬¥à 13.
Z			dx
	p		(4+ p3		)	:
		x		x

p p

•à¨¬¥-ïï ¯®¤áâ -®¢ªã x = t6, ¯®«ã稬 x = t3, 3 x = t2, dx = dt6 = 6t5 dt

¨

Z

6t5 dt

= 6

t2

dt = 6

Z

t2

+ 4 4

dt =

t3(4+ t2)

4 + t2

Z

t2 + 4

dt 24Z

t2 + 4 =

= 6Z

1 t2 + 4 dt = 6Z

4

dt

t

p6

p6

x

= 6t 12arctg

+ C = 6

x + 12arctg

+ C:

2

2

•à¨¬¥à 14.

Z

p

9 x2

dx:

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

7

•à¨¬¥-ïï ¯®¤áâ -®¢ªã x = 3sint, 3 6 x 6 3, 2 dx = d(3sint) = 3cost dt, ¯®«ã稬

Z	p		3cost dt = Z	p9cos2 t 3cost dt = 9Z
		9 9sin2 t

6 t 6 2, t = arcsin x3,

cos2 t dt =

= 9Z

2

dt =

2

Z dt + Z

cos2t dt = 2 t +

4 sin2t + C =

1 + cos2t

9

9

9

9

x

9

x

=

arcsin

+

sin

2arcsin

+ C:

2

3

4

3

•à¥®¡à §ã¥¬ ®â¤¥«ì-® ¢ëà ¦¥-¨¥ sin 2

x

, ¤«ï í⮣® ¢®á¯®«ì§ã¥¬áï

ä®à¬ã«®©:

arcsin 3

sin2t = 2sint cost = 2sint p1 sin2 t;

6 t 6

;

¯®í⮬ã

2

2

x

x

r1 sin2

x

sin 2arcsin

= 2sin arcsin

arcsin

=

3

3

3

= 2 3

r

1 9

= 9 x p9 x2:

Žâáî¤ ®ª®-ç ⥫ì-® ¯®«ã稬:

x

x2

2

Z

9

x

1

arcsin

9 x2 + C:

9 x2 dx = 2

3

+ 2 x

p

p

1.3. ˆ-⥣à¨à®¢ -¨¥ ¯® ç áâï¬

•ãáâì äã-ªæ¨¨ u(x), v(x) ¨¬¥îâ -¥¯à¥àë¢-ë¥ ¯à®¨§¢®¤-ë¥ u0(x), v0(x) - ¯à®¬¥¦ã⪥ (a; b), ⮣¤ á¯à ¢¥¤«¨¢® à ¢¥-á⢮:

Z	u(x)v0(x)dx = u(x)v(x) Z			v(x)u0(x)dx	(1)
¨«¨	Z	u dv = uv Z
			v du:

•¥®¡å®¤¨¬® § ¬¥â¨âì, çâ® ¯à¨¬¥-¥-¨¥ ¬¥â®¤ ¨-⥣à¨à®¢ -¨ï ¯® ç áâï¬ ¯à¨¢®¤¨â ª ç áâ¨ç-®¬ã ¨-⥣à¨à®¢ -¨î, â ª ª ª ¨ ¯à ¢ ï ç áâì ä®à¬ã«ë

(1) ᮤ¥а¦¨в ¨-в¥£а «. Ž¤- ª® ¯а¨ ¯а ¢¨«м-®¬ ¯а¨¬¥-¥-¨¨ ¬¥в®¤ ¨-в¥- £а « ¨§ ¯а ¢®© з бв¨ (1) ¡г¤¥в в ¡«¨з-л¬ ¨«¨ «¥£ª® б¢®¤пй¨¬бп ª в - ¡«¨з-®¬г. •а¨ ¢лз¨б«¥-¨¨ -¥ª®в®але ¨-в¥£а «®¢ ¬¥в®¤ ¨-в¥£а¨а®¢ -¨п ¯® з бвп¬ ¬®¦¥в ¯а¨¬¥-пвмбп -¥бª®«мª® а §. •а ¢¨«® ¨-в¥£а¨а®¢ -¨п ¯®

8	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

ç áâï¬ ¨¬¥¥â ¡®«¥¥ ®£à -¨ç¥--ãî ®¡« áâì ¯à¨¬¥-¥-¨ï, 祬 § ¬¥- ¯¥à¥- ¬¥--®©. •® ¥áâì æ¥«ë¥ ª« ááë ¨-â¥£à «®¢, - ¯à¨¬¥à:

Z

xk lnm x dx;

Z

xk sinax dx;

Z

xk cosax dx;

Z

xk eax dx;

Z

xk arcsinax dx;

Z

xk arccosax dx;

Z

xk arctgax dx

¨ ¤аг£¨¥, ª®в®ал¥ ¢лз¨б«повбп ¨¬¥--® б ¯®¬®ймо ¨-в¥£а¨а®¢ -¨п ¯® з - бвп¬.

I. •à¨ ¨-⥣à¨à®¢ -¨¨ äã-ªæ¨© ¢¨¤ Pm(x)sinax, Pm(x)cosax, Pm(x)eax,

£¤¥ Pm(x) ¯à®¨§¢®«ì-ë© ¬-®£®ç«¥- á⥯¥-¨ m, ¢ ª ç¥á⢥ äã-ªæ¨¨ u(x) ¨§ ä®à¬ã«ë (1) ¡¥à¥âáï ¬-®£®ç«¥- Pm(x), ¢ ª ç¥á⢥ v0(x) ¤à㣮© ᮬ-®¦¨-

⥫ì.

•à¨¬¥à 15. R x sin3x dx. •ãáâì u(x) = x, v0(x) = sin3x, ⮣¤

dv = sin3x dx = sin33 x d3x = 13 d cos3x:

Z	x sin3x dx = 3 Z							x d cos3x = 3					x cos3x Z								cos3x dx =
				1				1
		1		1				1					1
	=		x cos3x +			Z		cos3x dx =				x cos3x +							Z	cos3x d3x =
		3			3						3							9
													1								1
													=				x cos3x +						sin3x + C:
																3						9
•à¨¬¥à 16.				R xe 4x dx. •ãáâì u(x) = x, v0(x) = e 4x, ⮣¤
									e 4x				1
v0(x)dx = dv(x) = e 4x dx =										d( 4x) =					de 4x:
									4					4						e 4x dx =
	Z	xe 4x dx = 4 Z						x de 4x = 4 x e 4x Z
				1				1

=14 xe 4x + 14 Z e 4x dx = 14 xe 4x 161 Z e 4x d( 4x) =

=14 xe 4x 161 e 4x + C:

Šª ¡ë«® ᪠§ -® ¢ли¥, ¯а¨ ¢лз¨б«¥-¨¨ ¨-в¥£а «®¢ д®а¬г« ¨-в¥£а¨- а®¢ -¨п ¯® з бвп¬ ¬®¦¥в ¯а¨¬¥-пвмбп -¥бª®«мª® а §.

x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

9

•à¨¬¥à 17.

2

x

x2

v

0(

x

x

R

x

cos2x dx. •ãáâì u(1) =

,

) =1cos2

, ⮣¤

v0(x)dx = dv(x) = cos2x dx =

cos2x d2x =

d sin2x:

Z

2

2

x2 cos2x dx = 2 Z

x2 d sin2x = 2

x2 sin2x Z

sin2x dx2 : (2)

1

1

’ ª ª ª dx2 = 2x dx, â® ¢ ¯®á«¥¤-¥¬ ¨-â¥£à «¥ ¯®«ã稬

Z

Z

sin2x dx2 = (sin2x) 2x dx:

•®« £ ï u(x) = 2x, v0(x) = sin2x; v0(x)dx =

sin2x dx =

1 sin2x d2x =

= 21 d cos2x, ¨¬¥¥¬

2

Z

2x 2

d cos2x = Z

Z (sin2x) 2x dx =

x d cos2x =

1

= x cos2x Z

cos2x dx = x cos2x + 2 Z cos2x d2x =

1

1

= x cos2x +

sin2x + C:

2

Žª®-ç ⥫ì-®, ¯®¤áâ ¢«ïï ¯®á«¥¤-¨© १ã«ìâ â ¢ (2), ¯®«ã稬

Z

x2 cos2x dx = 2

x2 sin2x + x cos2x 2 sin2x + C =

1

1

=12 x2 sin2x + 12 x cos2x 14 sin2x + C:

II.•à¨ ¨-⥣à¨à®¢ -¨¨ äã-ªæ¨© ¢¨¤ : Pm(x)arcsinax, Pm(x)arccosax,

Pm(x)arctgax, Pm(x)lnax ¢ ª ç¥á⢥ v0(x) ¢ë¡¨à ¥âáï ¬-®£®ç«¥- Pm(x), ¢ ª ç¥á⢥ u(x) ®áâ ¢è ïáï äã-ªæ¨ï.

•à¨¬¥à 18. R x lnx dx. •ãáâì u(x) = lnx, v0(x) = x, ⮣¤

v0(x)dx = x dx = 12 dx2:

Z

x lnx dx = 2 Z

lnx dx2 = 2 x2 lnx Z

x2 d lnx =

1

1

1

1

1

x2

=

x2 lnx Z

x2

dx =

x2 lnx

+ C =

2

x

2

2

x2 lnx

x2

=

+ C:

2

4

+ 18 arctg2x + C: p

10	x1. •¥®¯à¥¤¥«¥--ë© ¨-â¥£à «...

•à¨¬¥à 19. R x arctg2x dx. •ãáâì u(x) = arctg2x, v0(x) = x, ⮣¤

				1																1
Z	x arctg2x dx =					Z		arctg2x dx2 =														x2 arctg2x
					2																2
	Z			x2 d arctg2x =									1				x2 arctg2x Z												2x2				dx =
													2															1 + (2x)2
		1						1										4x2		+ 1				1				1
	=			x2 arctg2x								Z											dx +				Z				dx =
		2									2							4x2		+ 1					2				4x2 + 1
		x2 arctg2x						1					1								d2x					x2 arctg2x					1
	=									x +						Z								=												x+
				2					4					8					(2x)2 + 1									2							4

III. •à¨ ¨-⥣à¨à®¢ -¨¨ äã-ªæ¨© eax sinbx, eax cosbx, ax2 + b ¨-⥣à¨- ஢ -¨¥ ¯® ç áâï¬ ¯à¨¬¥-ï¥âáï ¤¢ à § . ‚ëç¨á«¥-¨¥ ¨-â¥£à « ᢮¤¨âáï ª à¥è¥-¨î «£¥¡à ¨ç¥áª®£® ãà ¢-¥-¨ï ¯¥à¢®© á⥯¥-¨.

•à¨¬¥à 20.

Z

ex cosx dx = Z

cosx dex = ex cosx Z

ex d cosx =

= ex cosx + Z

ex sinx dx = ex cosx + Z

sinx dex =

= ex cosx + ex sinx Z

ex d sinx = ex cosx + ex sinx Z

ex cosx dx:

Ž¡®§- 稬 ç¥à¥§

I

= R

ex cosx dx. •¥à¥¯¨è¥¬ ¯®á«¥¤-¥¥ à ¢¥-á⢮ ¢ ¢¨¤¥:

I = ex cosx + ex sinx

I;

à¥è ï íâ® ãà ¢-¥-¨¥, ¯®«ã稬

2I = ex cosx + ex sinx;

I =

1

(ex cosx + ex sinx);

2

®âáî¤ ¨¬¥¥¬:

ex

Z

ex cosx dx =

(cosx + sinx) + C:

2

‡ ¬¥ç -¨¥. •à¨ ¨-⥣à¨à®¢ -¨¨ äã-ªæ¨© ¢¨¤ eax sinbx, eax cosbx ¨-- ⥣à¨à®¢ -¨¥ ¯® ç áâï¬ ¯à¨¬¥-ï¥âáï ¤¢ à § , ¯à¨ç¥¬ ¢ ®¡®¨å á«ãç ïå ¢ ª ç¥á⢥ ¬-®¦¨â¥«ï u(x) ¡¥à¥âáï äã-ªæ¨ï ®¤-®£® ¨ ⮣® ¦¥ ⨯ : ¯®ª § - ⥫ì- ï ¨«¨ âਣ®-®¬¥âà¨ç¥áª ï.