ÿþR +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
2 . n o d a r b +b a
N o d a r b +b a s s a t u r s : R a c i o n l a s d a <a s , +s t a s u n n e +s t a s . R a c i o n l a s
e l e m e n t r d a <a s , t o i n t e g r aa n a . *s t a s r a c i o n l a s d a <a s i z t e i k aa n a
e l e m e n t r d a <u s u m m a s v e i d .
2 . 1 . R a c i o n l a s d a <a s , +s t a s u n n e +s t a s
D e f i n +c i j a . P a r d a <v e i d a r a c i o n l u f u n k c i j u s a u c d i v u p o l i n o m u a t t i e c +b u
Q m ( x ) b 0 x m + b 1 x m - 1 + b 2 x m - 2 + K + b m - 1 x + b m
= ,
P n ( x ) a 0 x n + a 1 x n - 1 + a 2 x n - 2 + K + a n - 1 x + a n
k u r m , n " N . J a m < n , t a d ao f u n k c i j u s a u c p a r +s t u d a <v e i d a r a c i o n l u f u n k c i j u , j a
m e" n , p a r n e +s t u d a <v e i d a r a c i o n l u f u n k c i j u .
J a z e m i n t e g r <a f u n k c i j a i r n e +s t a d a <v e i d a r a c i o n l a f u n k c i j a , t a d t j i z s a k a k
p o l i n o m a u n +s t a s d a <v e i d a r a c i o n l a s f u n k c i j a s s u m m a , i z d a l o t s k a i t +t j a p o l i n o m u a r
x 4 - 3 x 3 + 2 x - 6
s a u c j a p o l i n o m u . P i e m r a m , f u n k c i j a i r n e +s t a d a <v e i d a r a c i o n l a
x 2 + 1
f u n k c i j a . I z d a l m p o l i n o m u s :
x 4 - 3 x 3 + 2 x - 6 x 2 + 1
x 4 + x 2 x 2 - 3 x - 1
- 3 x 3 - x 2 + 2 x - 6
- 3 x 3 - 3 x
- x 2 + 5 x - 6
- x 2 - 1
5 x - 5
T a d d o t o f u n k c i j u v a r p r r a k s t +t ad i :
x 4 - 3 x 3 + 2 x - 6 5 x - 5
= x 2 - 3 x - 1 + .
x 2 + 1 x 2 + 1
I n t e g r l i n o n e +s t a s d a <v e i d a r a c i o n l a s f u n k c i j a s s a d a l a d i v u i n t e g r <u s u m m , k u r
p i r m a j a m i n t e g r l i m z e m i n t e g r <a f u n k c i j a i r p o l i n o m s ( i n t e g r k a t r u s a s k a i t m o
a t s e v i a7i ) , o t r a j a m +s t a d a <v e i d a r a c i o n l a f u n k c i j a .
2 . n o d a r b +b a . 1 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
2 . 2 . R a c i o n l a s e l e m e n t r d a <a s . *s t a s r a c i o n l a s d a <a s i z t e i k aa n a
e l e m e n t r d a <u s u m m a s v e i d
L a i n o i n t e g r t u +s t u d a <v e i d a r a c i o n l u f u n k c i j u , s a u c j a p o l i n o m s j s a d a l a
r e i z i n t j o s . P o l i n o m a m i r i e s p j a m a s
e t r u v e i d u s a k n e s :
1 ) r e l a v i e n k r aa s a k n e a , t a i a t b i l s t r e i z i n t j s ( x - a ) ;
k
2 ) r e l a v a i r k k r t +g a s a k n e a a r k r t u k , t a i a t b i l s t r e i z i n t j s ( x - a ) ;
3 ) v i e n k r aa k o m p l e k s a s a k n e a + b i , t a d p o l i n o m a s a k n e i r a r + k o m p l e k s i s a i s t +t a i s
s k a i t l i s a - b i . `+m s a k n m a t b i l s t r e i z i n t j s ( x 2 + p x + q ) , k u r p = - 2 a ,
q = a 2 + b 2 ;
4 ) v a i r k k r t +g a k o m p l e k s a s a k n e a + b i a r k r t u k , a r + aa j g a d +j u m p o l i n o m a
s a k n e a r t o p a au k r t u i r a r + k o m p l e k s i s a i s t +t a i s s k a i t l i s a - b i . `+m s a k n m
k
a t b i l s t r e i z i n t j s ( x 2 + p x + q ) .
K a d s a u c j a p o l i n o m s s a d a l +t s r e i z i n t j o s , +s t a d a <v e i d a r a c i o n l a f u n k c i j a j s a d a l a
e l e m e n t r d a <s . A t k a r +b n o s a u c j a p o l i n o m a s a k n m i r
e t r u v e i d u e l e m e n t r d a <a s :
1 ) r e l a i v i e n k r aa i s a k n e i a a t b i l s t e l e m e n t r d a <a
A
;
x - a
2 ) r e l a i s a k n e i a a r k r t u k ( k > 1 ) a t b i l s t e l e m e n t r d a <a
A 1 A 2 A 3 A k
+ + + L + ;
k k - 1 k - 2
x - a
( x - a ) ( x - a ) ( x - a )
3 ) v i e n k r au k o m p l e k s u s a k Fu p r i m a + b i u n a - b i a t b i l s t e l e m e n t r d a <a
A x + B
;
x 2 + p x + q
4 ) k o m p l e k s u s a k Fu p r i m a + b i u n a - b i a r k r t u k ( k > 1 ) a t b i l s t e l e m e n t r d a <a
A 1 x + B 1 A 2 x + B 2 A k x + B k
+ + L + .
k k - 1
x 2 + p x + q
( x 2 + p x + q ) ( x 2 + p x + q )
A t b i l s t o ao e l e m e n t r d a <u s u m m u p i e v e d p i e k o p s a u c j a u n , p i e l +d z i n o t i e g kt o s k a i t +t j u
a r d o t s +s t s d a <v e i d a r a c i o n l s f u n k c i j a s s k a i t +t j u , a p r 7i n a n e z i n m o s k o e f i c i e n t u s .
1 5 - 1 1 x
P i e m r s . *s t u r a c i o n l u d a <v e i d a f u n k c i j u s a d a l +t e l e m e n t r d a <u s u m m .
x 3 - 4 x 2 + 3 x
R i s i n j u m s . D o t s f u n k c i j a s s a u c j u s a d a l +s i m r e i z i n t j o s :
2 . n o d a r b +b a . 2 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
1 5 - 1 1 x 1 5 - 1 1 x 1 5 - 1 1 x
= = .
x 3 - 4 x 2 + 3 x x ( x 2 - 4 x + 3 ) x ( x - 1 ) ( x - 3 )
T k v i s a s s a u c j a p o l i n o m a s a k n e s i r r e l a s u n v i e n k r aa s , k a t r a m r e i z i n t j a m s a u c j
a t b i l s t p i r m v e i d a e l e m e n t r d a <a :
1 5 - 1 1 x A B C A ( x - 1 ) ( x - 3 ) + B x ( x - 3 ) + C x ( x - 1 )
= + + = .
x ( x - 1 ) ( x - 3 ) x x - 1 x - 3 x ( x - 1 ) ( x - 3 )
T k p i r m u n p d j d a <a s i r v i e n d a s u n t o s a u c j i i r v i e n d i , t a d a r + s k a i t +t j i e m j b kt
v i e n d i e m :
1 5 - 1 1 x = A ( x - 1 ) ( x - 3 ) + B x ( x - 3 ) + C x ( x - 1 ) .
P d j a i v i e n d +b a i j i z p i l d s j e b k u r a m x , t p c v i e n d +b v a r i e v i e t o t j e b k u r a s x v r t +b a s .
P a t s v i e n k r ak a i s v e i d s , k a t r a s t k o e f i c i e n t u s A , B , C , i e v i e t o t x v i e t s a u c j a p o l i n o m a
s a k n e s :
x = 0 : 1 5 - 0 = A Å"( - 1 ) Å"( - 3 ) + B Å" 0 + C Å" 0 Ò! 1 5 = 3 A Ò! A = 5 ;
x = 1 : 1 5 - 1 1 = A Å" 0 + B Å"1 Å"( - 2 ) + C Å" 0 Ò! 4 = - 2 B Ò! B = - 2 ;
x = 3 : 1 5 - 3 3 = A Å" 0 + B Å" 0 + C Å" 3 Å" 2 Ò! - 1 8 = 6 C Ò! C = - 3 .
T t a d
1 5 - 1 1 x 5 2 3
= - - .
x ( x - 1 ) ( x - 3 ) x x - 1 x - 3
x 3 + 3 x 2 - 5 x + 1
P i e m r s . *s t u r a c i o n l u d a <v e i d a f u n k c i j u s a d a l +t e l e m e n t r d a <u
2
( x + 1 ) ( x 2 + 1 )
s u m m .
2
R i s i n j u m s . S a u c j s j a u i r s a d a l +t s r e i z i n t j o s . P i r m a j a m r e i z i n t j a m ( x + 1 ) a t b i l s t
r e l a d i v k r aa s a k n e x = - 1 , l +d z a r t o a t b i l s t o a e l e m e n t r d a <a i r o t r v e i d a ; r e i z i n t j a m
( x 2 + 1 ) a t b i l s t k o m p l e k s u v i e n k r au s k a i t <u p r i s x = ± i , t d j d i a t b i l s t o a
e l e m e n t r d a <a i r t r e a v e i d a . T t a d
2
x 3 + 3 x 2 - 5 x + 1 A B C x + D A ( x 2 + 1 ) + B ( x + 1 ) ( x 2 + 1 ) + ( C x + D ) ( x + 1 )
= + + =
2 2 2
x + 1 x 2 + 1
( x + 1 ) ( x 2 + 1 ) ( x + 1 ) ( x + 1 ) ( x 2 + 1 )
P i e l +d z i n s i m s k a i t +t j u s :
2
x 3 + 3 x 2 - 5 x + 1 = A ( x 2 + 1 ) + B ( x + 1 ) ( x 2 + 1 ) + ( C x + D ) ( x + 1 ) .
I e g kt a j v i e n d +b i e v i e t o s i m v i e n +g o r e l o s a k n i x = - 1 u n v l t r +s p a t v a <+g u s r e l u s
s k a i t <u s :
2 . n o d a r b +b a . 3 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
x = - 1 : - 1 + 3 + 5 + 1 = A Å" 2 + B Å" 0 + ( - C + D ) Å" 0 Ò! 8 = 2 A Ò! A = 4
x = 0 : 1 = A Å"1 + B Å"1 Å"1 + ( C Å" 0 + D ) Å"1 Ò! 1 = 4 + B + D Ò! B + D = - 3
x = 1 : 1 + 3 - 5 + 1 = A Å" 2 + B Å" 2 Å" 2 + ( C + D ) Å" 4 Ò! 0 = 8 + 4 B + 4 C + 4 D Ò!
Ò! B + C + D = - 2
x = 2 : 8 + 1 2 - 1 0 + 1 = A Å" 5 + B Å" 3 Å" 5 + ( 2 C + D ) Å"9 Ò! 1 1 = 2 0 + 1 5 B + 1 8 C + 9 D Ò!
Ò! 5 B + 6 C + 3 D = - 3
K o e f i c i e n t u B , C , D a p r 7i n aa n a i e s a m i e g u v u ai s i s t m u :
B + D = - 3
§#
ª#
B + C + D = - 2
¨#
ª#5 B + 6 C + 3 D = - 3
©#
N o p i r m v i e n d o j u m a i z t e i k s i m D = - 3 - B . I e v i e t o j o t t o o t r a j v i e n d o j u m , i e g ks i m
B + C - 3 - B = - 2 , n o k u r i e n e s C = 1 . I e v i e t o s i m t o t r e aa j v i e n d o j u m :
5 B + 6 + 3 ( - 3 - B ) = - 3 , 5 B + 6 - 9 - 3 B = - 3 , 2 B = 0 , B = 0 . T a d D = - 3 - B = - 3 .
V i s i k o e f i c i e n t i a p r 7i n t i , t t a d
x 3 + 3 x 2 - 5 x + 1 4 x - 3
= + .
2 2
x 2 + 1
( x + 1 ) ( x 2 + 1 ) ( x + 1 )
2 . 3 . R a c i o n l u e l e m e n t r d a <u i n t e g r aa n a
K a d +s t a r a c i o n l a f u n k c i j a i r s a d a l +t a e l e m e n t r d a <u s u m m , t a d k a t r u e l e m e n t r d a <u
n o i n t e g r a t s e v i a7i .
A p l kk o s i m , k i n t e g r k a t r a v e i d a e l e m e n t r d a <a s .
A d ( x - a )
1 ) d x = A = A l n x - a + C .
+"+"
x - a x - a
d x d ( x - 9 )
P i e m r s . = = l n x - 9 + C .
+"+"
x - 9 x - 9
- k + 1
A ( x - a ) A
- k
2 ) d x = A - a ) d ( x - a ) = A + C = + C .
+" k +"( x k - 1
- k + 1
( x - a ) ( 1 - k ) ( x - a )
- 2
d x ( x + 3 ) 1
- 3
P i e m r s . = + C = - + C .
+" 3 +"( x + 3 ) d ( x + 3 ) = 2
- 2
( x + 3 ) 2 ( x + 3 )
2 . n o d a r b +b a . 4 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
A x + B
3 ) d x = [ s a u c j u s a d a l +s i m p i l n a j o s k v a d r t o s u n i z m a n t o s i m s u b s t i t kc i j u ] =
+"
x 2 + p x + q
2
¡#
p p 2 ¤# A ›#t - p ž# + B
›# ž#
œ# Ÿ#
x 2 + p x + q = x + + q -
¢# œ# Ÿ# ¥#
2
# #
2 4
# #
¢# ¥#
= = d t =
+" 2
t + a 2
¢# p p p 2 ¥#
¢#t = x + 2 , x = t - , d x = d t , q - = a 2 ¥#
£# 2 4 ¦#
2
t d t A p d t A d ( t + a 2 ) + œ# B - A p d t
›# ž# ›# ž#
= A + B - Ÿ# Ÿ#
= =
œ#
+" 2 +" 2 +" 2 +" 2
t + a 2 2 t + a 2 2 t + a 2 2 t + a 2
# # # #
A A p 1 t
›# ž#
2
= l n t + a 2 + B - Ÿ#
Å" a r c t g + C =
œ#
2 2 a a
# #
A A p 1 x + p 2
›#B ž#
= l n x 2 + p x + q + - Ÿ#
Å" a r c t g + C .
œ#
2 2
# #
q - p 2 4 q - p 2 4
¡#x 2 8 x + 1 7 = x 2 2 Å" 4 x + 1 6 + 1 = - 4 ) 2 + 1 ¤#
( 6 x + 1 ) d x - - ( x
P i e m r s . = =
¢# ¥#
+"
x 2 - 8 x + 1 7
t = x - 4 , x = t + 4 , d x = d t
£# ¦#
2
6 ( t + 4 ) + 1 t d t d t 6 d ( t + 1 ) + 2 5 d t
= d t = 6 + 2 5 = =
+" 2 +" 2 +" 2 +" 2 +" 2
t + 1 t + 1 t + 1 2 t + 1 t + 1
2
= 3 l n t + 1 + 2 5 a r c t g t + C = 3 l n x 2 - 6 x + 1 7 + 2 5 a r c t g ( x - 4 ) + C .
A x + B
4 ) d x = [ i z m a n t o s i m 3 ) g a d +j u m l i e t o t o s u b s t i t kc i j u ] =
+" k
( x 2 + p x + q )
p
ž#
A ›#t - Ÿ#
+ B
œ#
2 t d t A p d t A p
›# ž# ›# ž#I .
# #
2
= d t = A + B - Ÿ# œ#
= A I + B - Ÿ#
œ#
k
+" k +" k +" k
2 2 2
2 2
# # # #
( t + a 2 ) ( t + a 2 ) ( t + a 2 )
2
K a t r u n o i n t e g r <i e m I u n I k a t r a d +s i m a t s e v i a7i :
- k + 1
2
t d t 1 - k 1 ( t + a 2 )
2 2
2
I = = ( t + a 2 ) d ( t + a 2 ) = Å" + C =
+" k +"
2
2 2 - k + 1
( t + a 2 )
1 1
= + C = + C .
k - 1 k - 1
2
2 ( 1 - k ) ( t + a 2 ) 2 ( 1 - k ) ( x 2 + p x + q )
2 . n o d a r b +b a . 5 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
2 2 2
d t 1 a 2 1 ( t + a 2 ) - t 1 t + a 2
I k = = d t = d t = d t -
+" k +" +" +"
2
a 2 2 + a 2 ) k a 2 2 + a 2 ) k a 2 2 + a 2 ) k
( t + a 2 ) ( t ( t ( t
2
1 t 1 d t 1 t Å" t 1 1
- d t = - d t = I k - I .
+" +" +"
a 2 2 + a 2 ) k a 2 2 + a 2 ) k - 1 a 2 2 + a 2 ) k a 2 - 1 a 2
( t ( t ( t
I n t e g r l i I a p r 7i n s i m , i n t e g r j o t p a r c i l i :
¡# ¤#
¢# ¥#
u = t d u = d t
¢# ¥#
¢# ¥#
t Å" t d t t d t
I = = d v = =
¢# ¥#
+" k k
2 2
( t + a 2 ) ( t + a 2 )
¢# ¥#
¢# t d t 1 ¥#
2
= I =
¢#v = k k - 1 ¥#
+"
2 2
( t + a 2 ) 2 ( 1 - k ) ( t + a 2 )
£# ¦#
t 1 d t
= - .
k - 1 +" k - 1
2 2
2 ( 1 - k )
2 ( 1 - k ) ( t + a 2 ) ( t + a 2 )
P d j a i s i n t e g r l i s s a k r +t a r I k - 1 . T a d i e g ks t a m
›# ž#
1 1 t 1
œ#
I k = I k - 1 - - I k - 1 Ÿ# =
a 2 a 2 œ# 2 ( 1 - k ) ( t 2 + a 2 ) k - 1 2 ( 1 - k ) Ÿ#
# #
›# 1 1 1 ž# t
= œ# + Å" Ÿ#I k - 1 -
œ# k - 1
2
a 2 a 2 2 ( 1 - k ) Ÿ# 2 a 2 ( 1 - k ) ( t + a 2 )
# #
j e b
2 k - 3 t
I k = I k - 1 + .
k - 1
2
2 a 2 ( k - 1 )
2 a 2 ( k - 1 ) ( t + a 2 )
P d j o f o r m u l u s a u c p a r r e k u r e n c e s f o r m u l u i n t e g r <a I k a t r a aa n a i . P c r e k u r e n c e s
f o r m u l a s s a m a z i n a i n t e g r <a k r t u , I k i z s a k o t a r I k - 1 , t o s a v u k r t i z s a k a a r I k - 2 , u t t . , l +d z
I 2 i z s a k a a r I 1 , k u r
d t d t 1 t
I 1 = = = a r c t g + C .
+" 1 +" 2
2
t + a 2 a a
( t + a 2 )
d x
P i e m r s . A t r a s t i n t e g r l i .
+" 3
( x 2 + 1 6 )
R i s i n j u m s . D o t a j g a d +j u m k = 3 , a = 4 . T a d p c f o r m u l a s i e g ks i m
2 . n o d a r b +b a . 6 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
2 Å" 3 - 3 x 3 x
I 3 = I 2 + = I 2 + ,
2 2
2 Å" 4 2 Å"( 3 - 1 ) 6 4
2 Å" 4 2 Å"( 3 - 1 ) ( x 2 + 1 6 ) 6 4 ( x 2 + 1 6 )
s a v u k r t p c t s p a aa s f o r m u l a s ( p i e k = 2 )
2 Å" 2 - 3 x 1 x
I 2 = I 1 + = I 1 + =
2 Å" 4 2 Å"( 2 - 1 ) 2 Å" 4 2 Å"( 2 - 1 ) ( x 2 + 1 6 ) 3 2 3 2 ( x 2 + 1 6 )
1 d x x 1 1 x x
= + = Å" a r c t g +
+"
3 2 x 2 + 1 6 3 2 ( x 2 + 1 6 ) 3 2 4 4 3 2 ( x 2 + 1 6 ) + C .
T d j d i
›#
d x 3 1 x x ž# x
= œ# + C =
+" 3 œ#1 2 8 a r c t g 4 + 3 2 ( x 2 + 1 6 ) Ÿ# + 2
Ÿ#
6 4
( x 2 + 1 6 ) # # 6 4 ( x 2 + 1 6 )
3 x 3 x x
= a r c t g + + C .
2
8 1 9 2 4 2 0 4 8 ( x 2 + 1 6 ) +
6 4 ( x 2 + 1 6 )
2 . 4 . R a c i o n l u f u n k c i j u i n t e g r aa n a ( a p k o p o j u m s )
V i s u i e p r i e k a u z r a k s t +t o p a r r a c i o n l u f u n k c i j u i n t e g r aa n u a p k o p o s i m , i z v e i d o j o t
Q m ( x ) d x ,
d a <v e i d a r a c i o n l a s f u n k c i j a s i n t e g r aa n a s s h m u . T t a d , l a i a t r a s t u i n t e g r l i
+"
P n ( x )
1 ) n o s a k a , v a i z e m i n t e g r <a f u n k c i j a i r +s t a d a <v e i d a r a c i o n l a f u n k c i j a . J a t n a v ( t . i .
m e" n ) , i z d a l a p o l i n o m u s u n a t d a l a v e s e l o d a <u :
Q m ( x ) R k ( x ) ;
= q m - n ( x ) +
P n ( x ) P n ( x )
2 ) s a u c j a p o l i n o m u s a d a l a r e i z i n t j o s ;
3 ) +s t u d a <v e i d a r a c i o n l u f u n k c i j u s a d a l a e l e m e n t r d a <u s u m m ;
4 ) n o i n t e g r p o l i n o m u q m - n ( x ) u n i e g kt s e l e m e n t r d a <a s .
3 x 4 - 1 4 x 3 + 2 0 x 2 + 1 5 x - 3 6
P i e m r s . A p r 7i n t d x .
+"
x 3 - 4 x 2 + 4 x
R i s i n j u m s . Z e m i n t e g r <a f u n k c i j a i r n e +s t a d a <v e i d a r a c i o n l a f u n k c i j a , j o s k a i t +t j a
p o l i n o m a p a k p e i r l i e l k a p a r s a u c j a p o l i n o m a p a k p i ( t . i . , 4 > 3 ) , t p c v i s p i r m s
j i z d a l a p o l i n o m i .
2 . n o d a r b +b a . 7 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
R +g a s T e h n i s k u n i v e r s i t t e . I n ~e n i e r m a t e m t i k a s k a t e d r a . L e k c i j u
k o n s p e k t s .
3 x 4 - 1 4 x 3 + 2 0 x 2 + 1 5 x - 3 6 x 3 - 4 x 2 + 4 x
3 x 4 - 1 2 x 3 + 1 2 x 2 3 x - 2
- 2 x 3 + 8 x 2 + 1 5 x - 3 6
- 2 x 3 + 8 x 2 - 8 x
2 3 x - 3 6
T t a d
3 x 4 - 1 4 x 3 + 2 0 x 2 + 1 5 x - 3 6 2 3 x - 3 6
= 3 x - 2 + .
x 3 - 4 x 2 + 4 x x 3 - 4 x 2 + 4
S a u c j a p o l i n o m u s a d a l +s i m r e i z i n t j o s u n +s t u r a c i o n l u d a <u s a d a l +s i m e l e m e n t r d a <u
s u m m :
2
2 3 x - 3 6 2 3 x - 3 6 2 3 x - 3 6 A B C A ( x - 2 ) + B x + C x ( x - 2 )
= = = + + =
2 2 2
x - 2 ) - 2
x
x 3 - 4 x 2 + 4 x ( x 2 - 4 x + 4 ) - 2 ) ( x x ( x - 2 )
x ( x
2
2 3 x - 3 6 = A ( x - 2 ) + B x + C x ( x - 2 )
x = 0 : - 3 6 = A Å" 4 + B Å" 0 + C Å" 0 Ò! - 3 6 = 4 A Ò! A = - 9
x = 2 : 4 6 - 3 6 = A Å" 0 + B Å" 2 + C Å" 0 Ò! 1 0 = 2 B Ò! B = 5
x = 1 : 2 3 - 3 6 = A Å"1 + B Å"1 + C Å"( - 1 ) Ò! - 1 3 = - 9 + 5 - C Ò! C = 9
T t a d
2 3 x - 3 6 9 5 9
= - + +
2
x
x 3 - 4 x 2 + 4 x - 2 ) - 2
( x
u n
›# ž#
3 x 4 - 1 4 x 3 + 2 0 x 2 + 1 5 x - 3 6 9 5 9
Ÿ#
d x = - 2 - + + d x =
+" +"œ#3 x x ( x - 2 ) 2
œ# Ÿ#
x 3 - 4 x 2 + 4 x x - 2
# #
d x d ( x - 2 ) x 2
- 2
= 3 x d x - 2 - 9 + 5 - 2 ) d ( x - 2 ) + 9 = 3 Å" - 2 x - 9 l n x +
+" +"d x +" +"( x +"
x x - 2 2
- 1
( x - 2 ) 3 5
+ 5 Å" + 9 l n x - 2 + C = x 2 - 2 x + 9 l n x - + 9 l n x - 2 + C .
- 1 2 x - 2
2 . n o d a r b +b a . 8 . l p p . A u g s t k m a t e m t i k a .
I . V o l o d k o
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