W y k Ba d X V I
T e m a t : C a Bk i n i e o z n a c z o n e
F u n k c j F ( x ) s p e Bn i a j c w a r u n e k F ( x ) = f ( x ) n a z y w a m y
f u n k c j p i e r w o t n . P o s z u k i w a n i e f u n k c j i p i e r w o t n e j
n a z y w a s i c a Bk o w a n i e m . C a Bk o w a n i e j e s t d z i a Ba n i e m
o d w r o t n y m d o r |n i c z k o w a n i a .
f ( x ) d x = F ( x ) + c ! ( F ( x ) + c ) 2 = f ( x )
+"
f u n k c j a
s t a Ba
p o d c a Bk o w a
s y m b o l f u n k c j a
c a Bk i p i e r w o t n a
W Ba s n o [c i c a Bk i n i e o z n a c z o n e j
W Ba s n o [ 1 . C z y n n i k s t a By m o |n a w y n i e [ p r z e d z n a k c a Bk i :
f ( x ) d x
+"c f ( x ) d x = c +"
W Ba s n o [ 2 . C a Bk a z s u m y ( r |n i c y ) d w c h f u n k c j i j e s t
r w n a s u m i e ( r |n i c y ) c a Be k z t y c h f u n k c j i :
f ( x ) d x g ( x ) d x
+"[ f ( x ) g ( x ) ] d x = +" +"
2
P o d s t a w o w e w z o r y c a Bk o w e
= 1
+"1 d x = x + c p o n i e w a | ( x + c ) 2
2
x n + 1
x n + 1 1
g d y |
x n d x = + c
+ c = "( n + 1 ) x n = x n
+"
n + 1
n + 1 n + 1
x r + 1
x r d x = + c d l a r `" - 1
+"
r + 1
1
d x = l n x + c
+"c o s d x = s i n x + c
+"
x
1
x
+"1 + x 2 d x = a r c t g x + c
+"e d x = e x + c
1
d x = a r c s i n x + c
+"s i n d x = - c o s x + c
+"
1 - x 2
3
P r z y k Ba d y :
( 2 x 3 - 3 x 2 + 5 x - 7 ) d x = 2 x 3 d x - 3 x 2 d x + 5 x d x - 7
+" +" +" +" +"1 d x =
x 4 x 3 x 2
= 2 + c 1 - 3 + c 2 + 5 + c 3 - 7 x + c 4 =
4 3 2
x 4 x 2
= - x 3 + 5 - 7 x + c
2 2
1 4 1 4
d x = d x
+
+" +" 4 x - 3 + x - 2 =
3
4 x 3 3 x 2
1 x - 2 4 x - 1 1 4
= " + " + c = - - + c
4 - 2 3 - 1
8 x 2 3 x
5 3
2 1
x 3 x 2 6
d x =
3
x 2 - 3 x x 2 - 2 x x + c
3
+"2 3 +"2 x - 3 x 2 = 2 - 3 + c = x 3
d x 5
5
2 4
3
M e t o d y c a Bk o w a n i a :
I . C a Bk o w a n i e p r z e z p o d s t a w i a n i e
2
f ( u ( x ) ) u ( x ) d x = f ( u ) d u = F ( u ) + c
+" +"
p o d s t a w i a m y :
P r z y k Ba d
2 x 2 + 7 = u
5
x ( 2 x 2 + 7 ) d x =
+"
4 x d x = d u
1
x d x = d u
4
6
1 1 u 6 1
= + c = ( 2 x 2 + 7 ) + c
+"u 5 d u =
4 4 6 2 4
5
I I . C a Bk o w a n i e p r z e z c z [c i
P o c h o d n a
i l o c z y n u
2 2
[ f ( x ) " g ( x ) ] 2 = f ( x ) " g ( x ) + f ( x ) " g ( x )
2 2
f ( x ) " g ( x ) d x + f ( x ) " g ( x ) d x
+"[ f ( x ) " g ( x ) ] 2 d x = +" +"
2 2
f ( x ) " g ( x ) = f ( x ) " g ( x ) d x + f ( x ) " g ( x ) d x
+" +"
c o d a j e n a m w z r n a c a Bk o w a n i e p r z e z c z [c i :
2 2
f ( x ) " g ( x ) d x = f ( x ) " g ( x ) - f ( x ) " g ( x ) d x
+" +"
l u b
2 2
f ( x ) " g ( x ) d x = f ( x ) " g ( x ) - f ( x ) " g ( x ) d x
+" +"
6
P r z y k Ba d 1 .
2
f ( x ) = 1 g ( x ) = l n x
+"l n x d x = +"1 "l n x d x =
1
f ( x ) =
+"1 d x = x g 2 ( x ) =
x
2 2
f ( x ) " g ( x ) d x = f ( x ) " g ( x ) - f ( x ) " g ( x ) d x
+" +"
1
= x l n x - x " d x = x l n x -
+" +"1 d x = x l n x - x + c
x
7
P r z y k Ba d 2 .
2
+"c o s x d x = +"c o s x "c o s x d x =
2
f ( x ) = c o s x g ( x ) = c o s x
2
f ( x ) = - s i n x g ( x ) =
+"c o s x d x = s i n x
= c o s x " s i n x + ( 1 - c o s 2 x ) d x
+"s i n x " s i n x d x = s i n x c o s x + +"
o s t a t e c z n i e o t r z y m u j e m y :
x d x = s i n x c o s x + x - x d x
+"c o s 2 +"c o s 2
2
2
+"c o s x d x = s i n x c o s x + x
s i n x c o s x + x
x d x = + c
+"c o s 2
8
2
Z a d a n i a n a w i c z e n i a
Z a d a n i a n a w i c z e n i a
1 6
1 . - x 2 d x
+"3 x 2 4 x + 5 - - + 8 3
x
x 3
2 6
4
2 . - + 8 x 3 + 5 d x
+"5 x 4 3 x 2 + 5 x - -
x
x 2
M e t o d a p r z e z p o d s t a w i e n i e :
2 x
2
4 . d x
3 . ( 2 x 3 + 3 ) d x
+"
+"3 x
3 x 2 + 5
2
3
5 . 6 .
+"2 x e 2 x + 3 d x +"8 x c o s ( x 4 + 3 ) d x
M e t o d a p r z e z c z [c i :
4 x
7 . 8 .
1 0 .
+"5 x c o s x d x +"7 x s i n x d x 9 .
+"5 x l n x d x +"2 x e d x
9
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