ÿþ
"
2 x + s i n 2 x d x
A 1 . Z b a d a z b i e |n o [ c a Bk i n i e w Ba [c i w e j k o r z y s t a j c z k r y t e r i u m i l o r a z o -
4 x 3 - x
2
"
"
2 x + 1 d x
"
w e g o , a z b i e |n o [ c a Bk i , z b a d a u |y w a j c k r y t e r i u m p o r ó w n a w c z e g o .
x 2 + 1
1
1
" g ( x ) = > 0 d l a x 2
2 x 2
2 x + s i n 2 x
f ( x )
4 x 3 - x
" k = l i m = l i m = 1 , 0 < k = 1 < "
x ’! " x ’! " 1
g ( x )
2 x 2
" "
2 x + s i n 2 x d x
d x
" z b i e |n a = Ò! z b i e |n a .
2 x 2 4 x 3 - x
2 2
" "
2 x + 1 2 x 1
" " "
" f ( x ) = > = = g ( x ) > 0 d l a x 1
x
x 2 + 1 x 2 + x 2
"
" "
2 x + 1 d x
d x
"
" " r o z b i e |n a d o " = Ò! r o z b i e |n a d o "
x
x 2 + 1
1 1
"
3
2 . P o d a [r o d e k , w s p ó Bc z y n n i k i o r a z w y z n a c z y p r z e d z i a B z b i e |n o [c i s z e r e g u ( 2 - 3 x ) n .
n
n = 1
" "
3 ( - 1 ) n 3 n + 1 2 n 2 ( - 1 ) n 3 n + 1
" ( 2 - 3 x ) n = x - = Ò! x 0 = , c n =
n n 3 3 n
n = 1 n = 1
3 n + 1 1
n
n
" l i m | c n | = l i m = 3 = Ò! R =
n ’! " n ’! "
n 3
"
1 ( - 1 ) n 3 n + 1 1 n " 3
" x = x 0 - R = , - = r o z b i e |n y d o "
3 n 3 n
n = 1 n = 1
"
( - 1 ) n 3 n + 1 1 n " 3 ( - 1 ) n
" x = x 0 + R = 1 , = n a p r z e m i e n n y z b i e |n y
n 3 n
n = 1 n = 1
1
" , 1 p r z e d z i a B z b i e |n o [c i
3
x 2
3 . F u n k c j f ( x ) = r o z w i n w s z e r e g M a c l a u r i n a , a n a s t p n i e w y z n a c z y f ( 4 ) ( 0 ) ,
3 x 2 - 2
f ( 5 ) ( 0 ) .
x 2 x 2
" f ( x ) = =
3
3 x 2 - 2
- 2 1 - x 2
2
"
3 n
" f ( x ) = - x 2 n + 2
2 n + 1
n = 0
3 2 2
" x 2 < 1 Ð!Ò! | x | < , R =
2 3 3
3 3
" c 4 = - : f ( 4 ) ( 0 ) = - 4 ! = - 1 8 , c 5 = 0 : f ( 5 ) ( 0 ) = 0 ,
2 2 2 2
1
4 . N a s z k i c o w a o b s z a r D o g r a n i c z o n y p r o s t y = 1 i w y k r e s e m f u n k c j i y = | l n x | , a n a s t p n i e
o b l i c z y j e g o p o l e .
y
1
x
1 e
1
e
"
e 1 e
" | D | = ( 1 - | l n x | ) d x = ( 1 + l n x ) d x + ( 1 - l n x ) d x
1 1
1
e e
" l n x d x = x ( l n x - 1 ) + C
1 e
1 e
1
" ( 1 + l n x ) d x = x l n x = ( 1 - l n x ) d x = x ( 2 - l n x ) = e - 2
1
1
e
e
1
1
e
1
" | D | = + e - 2 ( H" 1 . 0 8 6 1 )
e
"
"
( x + c o s x 2 ) d x
"
B 1 . K o r z y s t a j c z k r y t e r i u m i l o r a z o w e g o z b a d a z b i e |n o [ c a Bk i , a u |y -
x 2 - 1
2
"
( 2 x + 1 ) d x
w a j c k r y t e r i u m p o r ó w n a w c z e g o z b a d a z b i e |n o [ c a Bk i .
3 x 3 + x
1
1
"
" g ( x ) = > 0 d l a x 2
x
"
x + c o s x 2
"
f ( x )
x 2 - 1
" k = l i m = l i m = 1 , 0 < k = 1 < "
x ’! " x ’! " 1
g ( x )
"
x
"
" "
d x ( x + c o s x 2 ) d x
"
" " r o z b i e |n a d o " = Ò! r o z b i e |n a d o "
x
x 2 - 1
2 2
2 x + 1 2 x + x 1
" 0 < f ( x ) = < = = g ( x ) d l a x 1
3 x 3 + x 3 x 3 x 2
" "
d x ( 2 x + 1 ) d x
" z b i e |n a = Ò! z b i e |n a
x 2 3 x 3 + x
1 1
"
( 1 - 2 x ) n
2 . P o d a [r o d e k , w s p ó Bc z y n n i k i o r a z w y z n a c z y p r z e d z i a B z b i e |n o [c i s z e r e g u .
n 3 n
n = 1
"
( 1 - 2 x ) n " ( - 1 ) n 2 n 1 n 1 ( - 1 ) n 2 n
" = x - = Ò! x 0 = , c n =
n 3 n n 3 n 2 2 n 3 n
n = 1 n = 1
2
2 n 2 3
n
n
" l i m | c n | = l i m = = Ò! R =
n ’! " n ’! "
n 3 n 3 2
"
( - 1 ) n 2 n 3 n " 1
" x = x 0 - R = - 1 , - = r o z b i e |n y d o "
n 3 n 2 n
n = 1 n = 1
"
( - 1 ) n 2 n 3 n " ( - 1 ) n
" x = x 0 + R = 2 , = n a p r z e m i e n n y z b i e |n y
n 3 n 2 n
n = 1 n = 1
" ( - 1 , 2 ] p r z e d z i a B z b i e |n o [c i
x
3 . F u n k c j f ( x ) = r o z w i n w s z e r e g M a c l a u r i n a , a n a s t p n i e w y z n a c z y f ( 3 ) ( 0 ) ,
2 x 2 + 3
f ( 4 ) ( 0 ) .
x x
" f ( x ) = =
2
2 x 2 + 3
3 1 - - x 2
3
"
2 n
" f ( x ) = ( - 1 ) n x 2 n + 1
3 n + 1
n = 0
2 3 3
" - x 2 < 1 Ð!Ò! | x | < , R =
3 2 2
2 2 4
" c 3 = - : f ( 3 ) ( 0 ) = - 3 ! = - , c 4 = 0 : f ( 4 ) ( 0 ) = 0
3 2 3 2 3
À
4 . N a s z k i c o w a o b s z a r D o g r a n i c z o n y w y k r e s m f u n k c j i y = a r c c t g x , p r o s t y m i y = , y =
2
À
x , a n a s t p n i e o b l i c z y j e g o p o l e .
4
y
À
2
x
1 2
"
1 2
À À À
" | D | = - a r c c t g x d x + - x d x
2 2 4
0 1
1
" a r c c t g x d x = x a r c c t g x - l n x 2 + 1 + C
2
1
1
À À 1 À 1
" - a r c c t g x d x = x - x a r c c t g x + l n x 2 + 1 = + l n 2
2 2 2 4 2
0
0
2 2
À À À À À
" - x d x = x - x 2 =
2 4 2 8 8
1
1
3 1
" | D | = À + l n 2 ( H" 1 . 5 2 4 6 )
8 2
3
"
"
( x x - s i n x 2 ) d x
C 1 . K o r z y s t a j c z k r y t e r i u m i l o r a z o w e g o z b a d a z b i e |n o [ c a Bk i " , a u |y -
x 3 + x
2
"
( 2 x + 3 ) d x
w a j c k r y t e r i u m p o r ó w n a w c z e g o z b a d a z b i e |n o [ c a Bk i " .
x x + 1
1
1
"
" g ( x ) = > 0 d l a x 2
x x
"
x x - s i n x 2
"
f ( x )
x 3 + x
" k = l i m = l i m = 1 , 0 < k = 1 < "
x ’! " x ’! " 1
g ( x )
"
x x
"
" "
d x ( x x - s i n x 2 ) d x
" " z b i e |n a = Ò! " z b i e |n a
x x x 3 + x
2 2
2 x + 3 2 x 1
" f ( x ) = " > " " "
= = g ( x ) > 0 d l a x 1
x x + 1 x x + x x x
" "
d x ( 2 x + 3 ) d x
" " r o z b i e |n a d o " = Ò! " r o z b i e |n a d o "
x x x + 1
1 1
"
1
2 . P o d a [r o d e k , w s p ó Bc z y n n i k i o r a z w y z n a c z y p r z e d z i a B z b i e |n o [c i s z e r e g u ( 2 x + 4 ) n .
n 2
n = 1
" "
1 2 n 2 n
" ( 2 x + 4 ) n = ( x - ( - 2 ) ) n = Ò! x 0 = - 2 , c n =
n 2 n 2 n 2
n = 1 n = 1
2 n 1
n
n
" l i m | c n | = l i m = 2 = Ò! R =
n ’! " n ’! "
n 2 2
"
5 2 n 1 n " ( - 1 ) n
" x = x 0 - R = - , - = z b i e |n y
2 n 2 2 n 2
n = 1 n = 1
"
3 2 n 1 n " 1
" x = x 0 + R = - , = z b i e |n y
2 n 2 2 n 2
n = 1 n = 1
5 3
" - , - p r z e d z i a B z b i e |n o [c i
2 2
x 2
3 . F u n k c j f ( x ) = r o z w i n w s z e r e g M a c l a u r i n a , a n a s t p n i e w y z n a c z y f ( 4 ) ( 0 ) ,
4 - 2 x 2
f ( 5 ) ( 0 ) .
x 2 x 2
" f ( x ) = =
4 - 2 x 2 x 2
4 1 -
2
"
x 2 n + 2
" f ( x ) =
2 n + 2
n = 0
" "
x 2
" < 1 Ð!Ò! | x | < 2 , R = 2
2
1 4 !
" c 4 = : f ( 4 ) ( 0 ) = = 3 , c 5 = 0 : f ( 5 ) ( 0 ) = 0
2 3 2 3
4
4 . N a s z k i c o w a o b s z a r D o g r a n i c z o n y w y k r e s e m f u n k c j i y = l n x o r a z p r o s t y m i y = x - 1 ,
y = 1 , a n a s t p n i e o b l i c z y j e g o p o l e .
y
1
x
e
1 2
"
2 e
" | D | = ( ( x - 1 ) - l n x ) d x + ( 1 - l n x ) d x
1 2
" l n x d x = x ( l n x - 1 ) + C
2
2 e
e
x 2 3
" ( x - 1 - l n x ) d x = - x l n x = - 2 l n 2 ( 1 - l n x ) d x = x ( 2 - l n x ) =
2
2 2
1
1 2
2 l n 2 + e - 4
5
" | D | = e - ( H" 0 . 2 1 8 2 )
2
" "
( 2 x - c o s 2 x ) d x
"
D 1 . K o r z y s t a j c z k r y t e r i u m i l o r a z o w e g o z b a d a z b i e |n o [ c a Bk i , a u |y -
x x - 1
2
"
"
( x x + 1 ) d x
w a j c k r y t e r i u m p o r ó w n a w c z e g o z b a d a z b i e |n o [ c a Bk i " .
x 3 + 2 x
1
2
" g ( x ) = > 0 d l a x 2
x
"
2 x - c o s 2 x
"
f ( x )
x x - 1
" k = l i m = l i m = 1 , 0 < k = 1 < "
x ’! " x ’! " 2
g ( x )
x
"
" "
2 d x ( 2 x - c o s 2 x ) d x
"
" r o z b i e |n a d o " = Ò! r o z b i e |n a d o "
x x x - 1
2 2
" "
x x + 1 2 x x 2
"
" 0 < f ( x ) = " < = = g ( x ) d l a x 1
x 3 + 2 x x 3 x x
"
" "
2 d x ( x x + 1 ) d x
" " z b i e |n a = Ò! " z b i e |n a
x x x 3 + 2 x
1 1
"
3 x - 1 n
2 . P o d a [r o d e k , w s p ó Bc z y n n i k i o r a z w y z n a c z y p r z e d z i a B z b i e |n o [c i s z e r e g u n .
4
n = 1
"
3 x - 1 n " n 3 n 1 n 1 n 3 n
" n = x - = Ò! x 0 = , c n =
4 4 n 3 3 4 n
n = 1 n = 1
5
n 3 n 3 4
n
n
" l i m | c n | = l i m = = Ò! R =
n ’! " n ’! "
4 n 4 3
"
n 3 n 4 n "
" x = x 0 - R = - 1 , - = ( - 1 ) n n r o z b i e |n y
4 n 3
n = 1 n = 1
"
5 n 3 n 4 n "
" x = x 0 + R = , = n r o z z b i e |n y d o "
3 4 n 3
n = 1 n = 1
5
" - 1 , p r z e d z i a B z b i e |n o [c i
3
x
3 . F u n k c j f ( x ) = r o z w i n w s z e r e g M a c l a u r i n a , a n a s t p n i e w y z n a c z y f ( 3 ) ( 0 ) ,
4 x 2 + 3
f ( 4 ) ( 0 ) .
x x
" f ( x ) = =
4
3 + 4 x 2
3 1 - - x 2
3
"
4 n
" f ( x ) = ( - 1 ) n x 2 n + 1
3 n + 1
n = 0
" "
4 3 3
" - x 2 < 1 Ð!Ò! | x | < , R =
3 2 2
4 4 8
" c 3 = - : f ( 3 ) ( 0 ) = - 3 ! = - c 4 = 0 : f ( 4 ) ( 0 ) = 0
3 2 3 2 3
À
4 . N a s z k i c o w a o b s z a r D o g r a n i c z o n y w y k r e s e m f u n k c j i y = a r c t g x o r a z p r o s t y m i y = x ,
3
À
y = , a n a s t p n i e o b l i c z y j e g o p o l e .
3
y
À
3
"
x
1
3
"
"
3
1
À À
" | D | = x - a r c t g x d x + - a r c t g x d x
3 3
0 1
1
" a r c t g x d x = x a r c t g x - l n x 2 + 1 + C
2
1
1
À Àx 2 1 1 À
" x - a r c t g x d x = - x a r c t g x + l n x 2 + 1 = l n 2 -
3 6 2 2 1 2
0
0
"
"
3
3
À À 1 1 À
" - a r c t g x d x = x - x a r c t g x + l n x 2 + 1 = l n 2 -
3 3 2 2 1 2
1
1
À
" | D | = l n 2 - ( H" 0 . 1 6 9 5 )
6
6
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