matan-1_2

% 3 ) . $

	a2	(1 + cos 2t)dt =	a2		1
=				(t +		sin 2t) + C.
	2		2		2

L , 7

√

2

x

2

2

a2 − x2dx =

a

edJWTY

+

x

a − x

+ C,

2

a

2

sin 2t = sin t cos t =

x√

.

x2

sin t =

x

,

cos t = 1

,

1

a2 − x2

− a2

2

a

a2

C ,

7 7 8

dx = √x2

+ adx

√x2 + a

= x+ x2 + a = t, dt = 1 + √x2

+ a

dx

x

t

dt

=

= ln |t| + C = ln |x + x2 + a| + C.

t

#"L " 1 3 f (x) g(x)

∆ f (x)g(x)

!# ) '

f (x)g (x) ! !' &

E

	f (x)g (x)dx = f (x)g(x) −			f (x)g(x)dx,	(7.7)
		f (x)dg(x) = f (x)g(x) − g(x)df (x).			(7.8)
! D 7 f (x)g(x) 7

7 7

f (x)g (x) = (f (x)g(x))x − f (x)g(x).

C , M : M 7

, +, : ,

(f (x)g(x))xdx = f (x)g(x) + C,

, 7

f (x)g (x)dx = f (x)g(x) − f (x)g(x)dx + C.

% 3 ) . $

L C

f (x)g(x)dx,

, 7 N1 1O: N1 =O: f (x)dx = df (x): g (x)dx = dg(x)

7 N1 1O N 7 N1 =OO

7 7 7 : 7 <

8

x2 + adx =

f (x) = x2 + a, df (x) =

√x2

+ a, g(x) = x, dg(x) = dx =

x

2

x

dx

dx

= x x2 + a

−

= x

x2

+ a

−

x2

+ adx+a

.

√x2 + a

√x2 + a

$ 7 : , , 78

x2 + adx = 2

x2 + a + 2 ln |x +

x2 + a| + C.

x

a

C

In =

(x2 + a2)n , n N, a = 0.

dx

C 7 g(x) = x: f (x) = (x2 + a2)−n: df (x) = −2nx(x2 + a2)−n−1dx C M 7

I

=

x

+2n

x2dx

=

x

+2n

x2 + a2 − a2

dx

(x2 + a2)n

(x2 + a2)n+1

(x2 + a2)n

n

(x2 + a2)n+1

x

=

+ 2nIn − 2na2In+1.

(x2 + a2)n

:

In+1

=

x

+

2n − 1

In.

(7.9)

2na2(x2 + a2)n

2na2

C

7 n M 7 7 , In+1:

, , I1, I2, . . . , In 7: ,

I1 =

x2 + a2 = aedJVXa + C.

dx

1

x

% 4 ' ( & ,

L

I2 =	(x2 + a2)2	= 2a2(x2		+ a2) +		2a3 edJVXa + C.
	dx		x			1		x

+ 8 ; # ? > ; 0 A

LC "5"("'$" =- R(x) =	Pm (x)		Pm(x) Qn(x)
	Qn(x)	:

7 , 7 x m n : <

! M 7 7 E 7 <

R(x)dx,

R(x) Q

#"L " =- R(x) 0 # )

' F (x) R(x)

& #

5 M 7 7 7

E D : D <

7 7 ) 7

7 D 7

#"L " = 3 N 7 E O 1!

Pn(x) n

Pn(x) = a0(x − x1)α1 . . . (x − xk)αk (x2 + p1x + q1)β1 . . . (x2 + plx + ql)βl ,

xi 0 Pn(x) αi i = 1, . . . , k

x2 + pj x + qj j = 1, . . . , l !

α1 + · · · + αk + β1 + · · · + βl = n.

2

C : , 7 7					7 : 7 <
			Pm(x)		Q <
			Qn(x)
: , m < n L, :				<

Pm (x)	N : m ≥ nO 7
Qn(x)

% 4 ' ( & ,
	Pl (x)
S(x) +		: S(x) Q 7 , : l < n 5 : <
	Qn(x)
7
			βi		Mj x + Nj	(pj2 < 4qj ),
			(x − xi)αi		(x2 + pj x + qj )βj

αi, βj N: Q % , Q

%

#"L " = 2 1!

%

Pm(x)

k

αi

βij

l

βi

Mij x + Nij

i

Qn(x) =

(x

−

xj )j

+

(x2

+ pix + qi)j ,

i=1

j=1

=1

j=1

& & !

#

# 7 7 =-! 7 = 2

	R(x)dx =	Qn(x) dx
		Pm(x)

77

		S(x)dx		(8.1),
		(x − xi)j ,		(8.2)
		Aij dx
	(x2 + pix + qi)j dx			(8.3).
	Mij x + Nij

$ N=-O Q 7 , Q ,

, M 7 $ N= 3O Q , 8

(x − xi)j			Aij ln |x − xi|, j = 1.
	Aij dx			Aij	, j = 1,
		=		(1−j)(x−xi)j−1

$ N= 2O 7 77 <

, D 8

	(x2 + pix + qi)j dx =	2		(x2 + pix + qi)j dx+
	Mij x + Nij	Mij		2x + pi

% 4 ' ( & ,

Mij

2

+ Nij −

2

(x2 + pix + qi)j =

Mij pi

dx

d(x2 + pix + qi)

Mij pi

dx

+ Nij −

,

(x2 + pix + qi)j

2

((x + αi)2 + βi)j

pi

pi2

αi =

, βi = qi −

;

2

4

d(x2 + p x + q )

1

(x2 + p x + q )1−j , j = 1,

(x2 + pix + qi)j

ln |x2

+ pix + qi|, j = 1.

i i

=

1−j

i

i

$		(y2 + β)j , j ≥ 1,
Ij =
		dy

, 7 E N1 .O

! L 7 7 7 <

,

P (x)/Q(x)

N= 3O: N= 2O: 7 <

8

NTO N= 3O: N= 2O j = 1 7

7 N 7 O;

NTTO N= 3O j > 1

7 7: 7 7 , j −1; NTTTO N= 2O j > 1 , 7 7

N1 .O 77 7 <

7: 7 7 D, j − 1:

JkYWV	x2 + pix + qi .
	dx

! NTO: NTTO: NTTTO , : , 7 <

, P (x)/Q(x) C M 7 7 7

Q(x) = (x − x1)α1 . . . (x − xk)αk (x2 + p1x + q1)β1 . . . (x2 + plx + ql)βl .

: , 5 1 7 - &

% 4 ' ( & ,

# , M 77 <

D D : 7 D <

(x − x1)α1−1, . . . , (x − xk)αk −1,

(x2 + p1x + q1)β1−1, . . . , (x2 + plx + ql)βl−1.

, P (x)/Q(x) <

: , : P1(x)/Q1(x): <

7 7

Q1(x) = (x−x1)α1−1 . . . (x−xk)αk −1(x2+p1x+q1)β1−1 . . . (x2+plx+ql)βl−1.

C , 7 77 D E D :

D $ NTO NTTTO : , M 77

P2(x)/Q2(x): 7

Q2(x) = (x − x1) . . . (x − xk)(x2 + p1x + q1) . . . (x2 + plx + ql).

# 7 7 7 D 7 7 L

	Q(x)dx =	Q1(x) +	Q2(x)dx,
	P (x)	P1(x)	P2(x)

! 7 , Q1(x) Q2(x) E 7

, 7 , Q(x) <

7 D 7

5 : Q1(x) E 7 7 7

D 7 , Q(x) Q (x) 7 , 7

7 " :

, Q2(x) , Q(x)/Q1(x) 7 <

, 7 Q(x) Q1(x) F < 7F

L , 7 , P1(x) P2(x) C <

P1(x)/Q1(x) P2(x)/Q2(x) 7 : 7 , P1(x) P2(x) 7 , <

7 M 7 : , 7 Q1(x) Q2(x)

5 , D D M <

7 L < : 7 7

% 4 ' ( & ,

M D D x , <

D

L M : Q(x)

7 7

D Q(x)

C$" =- 7 L , 7

	6 7x x2
	x4 − 2x3−+ 3x−2 − 2x + 1dx.

$7 7

Q(x) = x4 − 2x3 + 3x2 − 2x + 1, Q (x) = 4x3 − 6x2 + 6x − 2.

' E M D 7 ,

Q1(x) = x2 − x + 1.

C Q(x) Q1(x) F 7F: 7

Q2(x) = x2 − x + 1.

P1(x) P2(x) 7 7 , <

7 M 7 : 7 L 7

	6 7x x2		Ax + B		Cx + D
	− −	dx =		+		dx.
	x4 − 2x3 + 3x2 − 2x + 1		x2 − x + 1		x2 − x + 1

C 7 M 7 8

		6 − 7x − x2	=
		x4 − 2x3 + 3x2 − 2x + 1
=	A(x2 − x + 1) − (Ax + B)(2x − 1)		+	Cx + D	.
		x2 − x + 1		x2 − x + 1

7 7 7 :

, 7 , C , 7

6 −7x −x2 = A(x2 −x + 1) −(Ax + B)(2x −1) + (Cx + D)(x2 −x + 1).

% 5 ( ' ( &

M x0, x1, x2 x3: , 7 7 <

A + D

C = 1,

C = 0,

−2B D−+ C =− 7,

− −

:

−

:

# 7

E M 7 : 7

A = 2

:

= 0

D = 1

B = 3

A + B + D = 6.

C

7 7 L 7

6

7x

x2

x + 3

dx

− −

dx =

2

+

.

x4 − 2x3 + 3x2 − 2x + 1

x2 − x + 1

x2 − x + 1

! , 7 , 8

√3

√3

x2 − x + 1

(x − 1/2)2 + 3/4

dx

=

dx

=

2

edJVX

2x − 1

+ C.

L , 7 7

6 7x

x2

dx =

2x + 3

+

2

edJVX

2x − 1

+ C.

x4 − 2x3−+ 3x−2 − 2x + 1

x2 − x + 1

√3

√3

+ )

? ; # ? > ;

7 7 <

D $

	R x,	ax + b		r1	, . . . ,	ax + b		rn	dx,	(9.1)
		cx + d				cx + d

rk Q: k = 1, . . . , n: a, b, c, d R: ad − bc = 0:

ax + b = tp, cx + d

p Q 7 E 7 r1, r2, . . . , rn: <

5 : D < 7

x =	dtp − b	,	dx = tp−1	p(ad − bc)	dt.	(9.2)
	ctp − a
				(ctp − a)2

C N. 3O N.-O: , 7 7

% 5 ( ' ( &

$

	R(x, ax2		+ bx + c)dx, a = 0, b2 − 4ac = 0,

7 7 D 7

8

= ±t ± √ax,

a > 0,

ax

2

+ bx + c

ax2

2

+ bx + c =

t(x± x±),

b2

4ac > 0,

ax + bx + c = tx

√c, c > 0,

x1 Q

± − 1

−

ax + bx + c

2

D,

$

xm(axn + b)pdx,

(9.3)

a, b R: m, n, p Q: , 7 a, b, n, p = 0:

$ N. 2O

D D , D8

p Z Q x = tq , q Q 7 m, n;

mn+1 Z Q axn + b = tq, q Q 7 p;

p + mn+1 Z Q a + bx−n = tq, q Q 7 p. L 7 7: , M , ' : E

[*[ 7 7 C ( E

: , N. 2O D , D , M 7

$

R(sin x, cos x)dx

7 7

t = VXx2 :

sin x =	2t	,		cos x =	1 − t2	,	dx =	2dt	.
	1 + t2				1 + t2			1 + t2
$				R(Wgx, Jgx)dx

-0 , 3 & >$ . 1

! # 1 4 > &

% 5 ( ' ( &

7 7 t = Vgx2 :
M 7		2t			2		2dt
Wgx =				, Jgx =	1 + t	, dx =			.
		− t	2		2		1 − t	2
1					1 − t