background image

Zadanie 1. 

 

( )

x

x

x

Ce

y

Ce

y

C

e

y

C

x

y

dx

dy

y

dx

dy

y

y

dx

dy

y

dx

dy

x

x

y

dx

dy

3

0

3

3

2

ln

ln

ln

ln

ln

3

ln

3

1

3

1

3

0

3

5

3

=

=

+

=

+

=

=

=

=

=

=

 

 
 
 

(

)

27

13

9

13

3

1

27

13

9

13

3

1

27

13

,

9

13

,

3

1

0

3

5

3

2

1

3

5

3

3

3

2

5

3

2

2

2

3

0

2

2

2

2

2

2

+

+

=

+

=

+

+

=

=

=

=

=

=

=

=

+

=

+

+

+

+

=

+

+

=

x

x

Ce

y

y

y

y

x

x

y

c

b

a

c

b

b

a

a

x

x

c

bx

ax

b

ax

x

x

c

bx

ax

b

ax

b

ax

dx

dy

c

bx

ax

y

x

sz

sz

sz

 

Zadanie 2. 

 

(

)

(

)

(

)

(

)

1

1

1

:

1

1

ln

1

ln

ln

ln

1

ln

1

ln

1

1

1

1

1

1

1

1

1

1

2

2

1

2

1

1

1

1

)

1

2

(

1

1

1

1

1

1

1

1

)

1

2

(

2

2

2

2

2

2

2

2

2

2

2

2

2

+

+

=

+

=

+

=

=

=

+

=

+

=

=

=

=

=

+

+

+

+

+

=

+

+

+

+

+

+

=

=

+

=

+

=

=

+

+

+

+

=

x

x

x

x

x

sz

sz

Ce

x

y

u

x

y

ie

podstawien

Ce

u

Ce

u

Ce

u

C

e

u

C

x

u

dx

du

u

dx

du

u

dx

u

du

u

u

dx

du

u

x

x

u

x

u

x

x

u

u

x

x

dx

du

u

x

x

u

x

x

u

x

dx

du

u

dx

du

u

dx

dy

u

x

y

u

y

y

x

y

x

x

y

x

y

dx

dy

 

 

 

background image

 

Zadanie 3. 

 

(

)

(

)

(

)

(

) (

)

(

) (

)

(

)

(

)

y

x

y

x

C

x

y

x

y

x

C

x

y

x

y

x

C

C

x

y

x

y

x

y

x

z

ie

podstawien

C

x

z

z

dx

dz

z

dz

dx

dz

z

dz

z

z

dx

dz

z

z

zdx

dz

z

z

z

dx

dz

dx

dz

dx

dz

z

z

dx

dz

z

dx

dz

z

z

dx

dz

z

z

dx

dz

dx

dy

x

z

y

y

x

z

dx

dy

y

x

y

x

+

+

+

=

+

+

+

=

+

+

+

=

+

=

+

+

+

+

=

+

=

+

=

+

=

+

=

+

=

+

=

+

=

+

+

=

+

+

+

=

+

+

+

=

=

+

=

=

+

+

+

+

+

ln

2

ln

2

2

ln

2

ln

2

:

ln

2

1

2

1

2

1

2

1

2

1

2

0

2

0

1

2

2

1

0

1

1

2

1

1

0

)

1

2

2

(

1

 

Zadanie 4. 

 

( )

( )

( )

( )

( )

x

Cx

y

y

y

y

x

x

x

y

x

u

xdx

du

dx

du

x

x

x

x

u

x

x

u

dx

du

x

x

x

u

dx

du

x

dx

dy

x

x

u

y

Cx

y

Cx

y

C

x

y

C

x

y

dx

x

dy

y

dx

x

dy

y

x

y

dx

dy

x

y

dx

dy

x

dx

dy

x

y

x

sz

sz

2

1

2

1

2

1

2

1

1

1

1

1

1

1

1

1

ln

ln

ln

ln

ln

ln

ln

1

1

1

1

1

:

0

1

0

1

2

2

2

2

1

1

0

1

1

=

+

=

=

=

=

=

=

=

+

=

=

=

=

+

=

+

=

=

=

=

=

+

=

+

+