ÿþS I M R 2 0 1 0 / 1 1 , A n a l i z a 1 , w y k Ba d 9 , 2 0 1 0 - 1 2 - 0 2
C a Bk a n i e o z n a c z o n a
D e f i n i c j a : N i e c h d a n a b d z i e f u n k c j a f : I ’! R , g d i z e I ‚" R j e s t p r z e d z i a Be m . F u n k c j
p i e r w o t n t e j f u n k c j i n a z y w a m y k a |d f u n k c j F : I ’! R t a k , |e ( "x " I ) F ( x ) = f ( x ) .
Z b i ó r w s z y s t k i c h f u n k c j i p i e r w o t n y c h f u n k c j i f n a z y w a m y c a Bk n i e o z n a c z o n i o z n a c z a m y
s y m b o l e m :
f ( x ) d x
U w a g a 1 : P r z e d z i a B I m o |e b y d o m k n i t y I = < a , b > . W t e d y w k o Dc a c h p r z e d z i a Bu
F ( a ) i F ( b ) s p o c h o d n y m i j e d n o s t r o n n y m i .
U w a g a 2 : J e |e l i i s t n i e j e f u n k c j a p i e r w o t n a f u n k c j i f , t o i s t n i e j e n i e s k o Dc z e n i e w i e l e f u n k c j i
p i e r w o t n y c h r ó |n i c y c h s i t y l k o o s t a B:
J e |e l i F 1 ( x ) = f ( x ) i F 2 ( x ) = f ( x ) t o ( F 1 ( x ) - F 2 ( x ) ) = 0 c z y l i F 1 ( x ) - F 2 ( x ) = C n a
p r z e d z i a l e I .
U w a g a 3 : O p e r a c j o b l i c z a n i a c a Bk i n i e o z n a c z o n e j n a z y w a m y c a Bk o w a n i e m . C a Bk o w a n i e j e s t
o p e r a c j o d w r o t n d o r ó |n i c z k o w a n i a . W i k s z o [ t e c h n i k o b l i c z a n i a p o c h o d n y c h m a s w o j e
o d p o w i e d n i k i o b l i c z a n i a c a Be k .
C a Bk i n i e o z n a c z o n e f u n k c j i e l e m e n t a r n u c h
x ±+ 1
1 . x ±d x = + C , ± = - 1
± + 1
1
2 . d x = l n | x | + C
x
3 . e x d x = e x + C
4 . s i n x d x = - c o s x + C
5 . c o s x d x = s i n x + C
1
6 . d x = t g x + C
c o s 2 x
1
7 . d x = - c t g x + C
s i n 2 x
1
8 . d x = a r c t g x + C
1 + x 2
1
"
9 . d x = a r c s i n x + C
1 - x 2
1 0 . s i n h x d x = c o s h x + C
1 1 . c o s h x d x = s i n h x + C
1
1 2 . d x = t g h x + C
c o s h 2 x
1
1
1 3 . d x = - c t g h x + C
s i n h 2 x
P o d s t a w o w e w Ba s n o [c i c a Bk i n i e o z n a c z o n e j
Z a k Ba d a m y , |e f u n k c j e f , g : I ’! R s c a Bk o w a l n e . W t e d y :
1 . a f ( x ) d x = a f ( x ) d x
2 . ( f ( x ) + g ( x ) ) d x = f ( x ) d x + g ( x ) d x
3 . ( f ( x ) - g ( x ) ) d x = f ( x ) d x - g ( x ) d x
P r z y k Ba d y :
x 4 x 3
1 2 x 3 - 6 x 2 + 4 x - 5 d x = 1 2 x 3 d x - 6 x 2 d x + 4 x 1 d x - 5 x 0 d x = 1 2 - 6 +
4 3
x 2 x 1
4 - 5 + C = 3 x 4 - 2 x 3 + 2 x 2 - 5 x + C
2 1
"
3 4 2 1 1
2 2
"
2 x + 6 x + - + d x = 2 x 1 d x + 6 x d x + 3 x - d x - 4 x - 1 d x + 2 x - 2 d x =
x x x 2
3 1
2 2 " "
x x x - 1 2
x 2 + 6 + 3 - 4 l n | x | + 2 + C = x 2 + 4 x x + 6 x - 4 l n | x | - + C
3 1
- 1 x
2 2
s i n 2 x 1 - c o s 2 x 1 1
d x = d x = - 1 d x = d x - d x = t g x - x + C
c o s 2 x c o s 2 x c o s 2 x c o s 2 x
x 2 x 2 + 1 - 1 1
d x = d x = d x - d x = x - a r c t g x + C
x 2 + 1 x 2 + 1 x 2 + 1
C a Bk o w a n i e p r z e z p o d s t a w i e n i e
J e [l i g : I 1 ’! I 2 j e s t r ó |n i c z k o w a l n a , f : I 2 ’! R , i F : I 2 ’! R j e s t f u k c j p i e r w o t n f
( ( "t " I 2 ) F ( t ) = f ( t ) ) t o i s t n i e j e p o n i |s z a c a Bk a :
f ( g ( x ) ) · g ( x ) d x = F ( g ( x ) ) + C
U w a g a : T e c h n i k a t a j e s t o d p o w i e d n i k i e m o b l i c z a n i a p o c h o d n e j f u n k c j i z Bo |o n e j .
P r z y k Ba d :
x
d x
x 2 + 5
1
t = x 2 + 5
P o d s t a w i a m y : , s t d x d x = d t
d t = 2 x d x
2
x 1 1 1
d x = d t = l n | t | + C
x 2 + 5 2 t 2
W r a c a m y d o z m i e n n e j x
x 1
d x = l n | x 2 + 5 | + C
x 2 + 5 2
1
U w a g a : W t y m p r z y k Ba d z i e t = g ( x ) = x 2 + 5 , g ( x ) = 2 x , f ( t ) =
2 t
P r z y k Ba d y :
ñø üø
ôø
t = x 2 ôø
òø ýø
2 1 1 1 2
x e x d x = d t = 2 x d x = e t d t = e t + C = e x + C
ôø ôø
2 2 2
óø þø
1
x d x = d t
2
2
ñø üø
ôø
t = c o s x ôø
òø ýø
s i n x - 1
d x = d t = - s i n x d x = d t = - a r c t g t + C = - a r c t g ( c o s x ) + C
ôø ôø
c o s 2 x + 1 t 2 + 1
óø þø
s i n x d x = - d t
e x 1
t = e x
" "
d x = = d t = a r c s i n t + C = a r c s i n ( e x ) + C
d t = e x d x
1 - e 2 x 1 - t 2
ñø üø
ôø
t = x 2 ôø
1
òø ýø
x d t 1 1
2
d x = d t = 2 x d x = d t = a r c t g t + C = a r c t g ( x 2 ) + C
ôø ôø
x 4 + 1 t 2 + 1 2 2
óø þø
1
x d x = d t
2
ñø üø
3
ôø
t = x 2 + 1 ôø
òø ýø
"
2
1 " 1 t 1 " 3
x x 2 + 1 d x = d t = 2 x d x = t d t = + C = x 2 + 1 + C
3
ôø ôø
2 2 3
óø þø
1
2
x d x = d t
2
ñø üø
òø t = l n x ýø
l n 3 x 1 1
d x = d x = t 3 d t = t 4 + C = l n 4 x + C
óø þø
x d t = 4 4
x
ñø üø
ôø
s = t 2 ôø
òø ýø
s i n x c o s x t 1 1 1
t = s i n x
d x = = d t = d t = 2 s d s = d s = l n | s | +
ôø ôø
d t = c o s x d x
s i n 2 x + 4 t 2 + 4 2 s 2
óø þø
1
s d s = d t
2
1 1
C = l n | t 2 + 4 | + C = l n | s i n 2 x + 4 | + C
2 2
P o d s t a w i e n i e l i n i o w e : t = a x + b
1
J e |e l i F ( t ) = f ( t ) o r a z a , b " R , a = 0 t o f ( a x + b ) d x = F ( a x + b ) + C
a
P r z y k Ba d y :
1 1
d x = { t = 2 x + 7 } = l n | 2 x + 7 | + C
2 x + 7 2
1
s i n ( 4 x - 1 ) d x = { t = 4 x - 1 } = - c o s ( 4 x - 1 ) + C
4
1
e - 2 x d x = { t = - 2 x } = - e - 2 x + C
2
1 1 1 1 x 1 x
d x = d x = d x = { t = } = a r c t g ( ) + C
x 2
x 2 + 4 x 2 4 2 2 2
+ 1
4 ( + 1 )
2
4
1 1 1 1 1
d x = d x = d x = d x =
x 2 + 4 x + 1 3 ( x + 2 ) 2 + 9 ( x + 2 ) 2 9 x + 2 2
9 ( + 1 ) + 1
9 3
x + 2 1 x + 2
{ t = } = a r c t g ( ) + C
3 3 3
P o d s t a w i e n i e z a m i a n o w n i k
J e |e l i l i c z n i k f u n k c j i p o d c a Bk o w e j j e s t p o c h o d n m i a n o w n i k a p o m n o |o n p r z e z s t a B, t o
p o d s t a w i a j c n o w z m i e n n z a m i a n o w n i k m a m y :
a f ( x ) a
t = f ( x )
d x = = d t = a l n | t | + C = a l n | f ( x ) | + C
d t = f ( x ) d x
f ( x ) t
P r z y k Ba d y :
ñø üø
ôø
t = c o s x ôø
òø ýø
s i n x - 1
d x = d t = - s i n x d x = d t = - l n | t | + C = - l n | c o s x | + C
ôø ôø
c o s x t
óø þø
s i n x d x = - d t
ñø üø
ôø
t = x 3 + 4 x ôø
òø ýø
6 x 2 + 8 2
d x = d t = ( 3 x 2 + 4 ) d x = d t = 2 l n | t | + C = 2 l n | x 3 + 4 x | + C
ôø ôø
x 3 + 4 x t
óø þø
d x = 2 d t
e x 1
t = e x - 2
d x = = d t = l n | t | + C = 2 l n | e x - 2 | + C
d t = e x d x
e x - 2 t
3
ñø üø
òø t = l n x ýø
1 1
d x = 1 = d t = l n | t | + C = 2 l n | l n x | + C
óø þø
x l n x d t = d x t
x
C a Bk o w a n i e p r z e z c z [c i
J e [l i I j e s t p r z e d z i a Be m , f , g : I ’! R s r ó |n i c z k o w a l n e o r a z f u n k c j a f g j e s t c a Bk o w a l n a t o :
f ( x ) · g ( x ) d x = f ( x ) · g ( x ) - f ( x ) · g ( x ) d x
U w a g a : T e c h n i k a t a j e s t o d p o w i e d n i k i e m o b l i c z a n i a p o c h o d n e j i l o c z y n u f u n k c j i .
P r z y k Ba d :
x l n x d x
f ( x ) = l n x g ( x ) = x
C a Bk u j e m y p r z e z c z [c i :
1 x 2
f ( x ) = g ( x ) = x d x =
x 2
x 2 1 x 2 x 2
x l n x d x = l n x · - · x d x = l n x - x d x = l n x - x + C
2 x 2 2
P r z y k Ba d y :
f ( x ) = x g ( x ) = s i n x
x s i n x d x = = - x c o s x - - c o s x x d x = - x c o s x + s i n x +
f ( x ) = 1 g ( x ) = - c o s x
C
f ( x ) = e x g ( x ) = s i n x
e x s i n x d x = = - e x c o s x - - e x c o s x d x = - e x c o s x +
f ( x ) = e x g ( x ) = - c o s x
e x c o s x d x
O b l i c z a m y :
f ( x ) = e x g ( x ) = c o s x
e x c o s x d x = = e x s i n x - e x s i n x d x
f ( x ) = e x g ( x ) = s i n x
S t d m a m y :
e x s i n x d x = - e x c o s x + e x s i n x - e x s i n x d x
2 e x s i n x d x = e x ( s i n x - c o s x ) + C
1
e x s i n x d x = e x ( s i n x - c o s x ) + C
2
ñø üø
f ( x ) = a r c t g x g ( x ) = 1
òø ýø
x
1
a r c t g x d x = = x a r c t g x - d x = { t = 1 +
óø þø
f ( x ) = g ( x ) = x
1 + x 2
1 + x 2
1 1
x 2 , d t = 2 x d x } = x a r c t g x - d t = x a r c t g x - l n | 1 + x 2 | + C
2 t 2
4
Wyszukiwarka
Podobne podstrony:
SIMRAn1Wyk2SIMRan1kol2 przyg2013SIMRAn1Wyk6SIMRAn1Wyk7SIMRAnZespZad4SIMRAnZespWyk2SIMRan1kol1 przyg2013SIMRAnZespWyk1SIMRAnZespWyk3SIMRAn1Wyk5SIMRAn1Wyk4SIMRAn1Wyk3SIMRAn1Wyk3SIMRAn1Wyk10więcej podobnych podstron