3 Aporte 199 Trabajo Fase 2

7/21/2019 3 Aporte 199 Trabajo Fase 2

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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA - UNADEscuela de Ciencias Bsicas, Tecnologa e Ingeniea

100411- Clculo in!egalTrabajo Colaborativo Fase 2

"U#A DE ACTIVIDADES

$ASE %& TRABA'O COLABORATIVO %

())*(( + CALCULO INTE"RAL

RESENTADO OR

NATALY CARDONA VALDES

C.DI"O9022!10"#0

"RUO200!11$119

RESENTADO AL IN"ENIERO

L%&S FERNANDO AR&AS RA'&RE( )T*tor+

UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA

ESCUELA DE CIENCIAS B/SICAS, TECNOLO"#A E IN"ENIER#A

0AR1O DE %)(2


2/23



E3ecicio *

2

5

4+x

x24dx

Sol*,i./

Cal,*lar la i.teral i.ei.ia/

2

5

4+x

x24dx=21+4 ln( 5+212 )

4+xx24

dx=4 ln(1 4x2x2

+x

2)+x24+C

4+xx24

dx

A3li,ar la rela e la s*a/ f(x )g (x )dx= f(x )dx g (x )dx

4 1x24

dx+ xx24

dx

4+xx

24dx=4 ln(1 4x2x

2 +

x

2)

4+xx24

dx

Toar la salia e *.a ,o.ta.te/ af(x )dx=a f(x )dx


3/23



4 1x

24dx

5ara b x2a s*stit*6e.o x=

a

b sec (u )

A3li,ar la s*stit*,i. e *.a i.teral/ f(g (x ))g'(x )dx= f(u)du 7 u=g (x )

x=2 sec (u )

dx=

2

cos (u )tan

(u ) du

4 1(2sec (u))

2

4

2

cos (u )tan (u )du

4

2

cos (u)tan (u)

4 sec2 (u )4

du


42

1

cos (u )tan (u )

4 sec2 (u )4

du

Reei.ir

42tan (u)cos (u )

4 sec2 (u )4

du

4 sec2 (u )4=4 (4 sec

2 (u )4

1)


4/23



42

tan (u)cos (u)

4

(4sec

2 (u

)4 1)

du

Reei.ir

42

tan (u )cos (u )

2sec2 (u )1

du

%tili8ar la si*ie.te ie.tia/sec

2 (x )=1 ta n

2 (x )

tan (u )cos (u )

21+1tan2 (u)

du


42( 12 )tan (u )cos (u )

1+1tan2 (u )

du

42( 12 )tan (u)cos (u)

tan2 (u )

du

tan2 (u )=( tan (u )) 7 as*ie.o tan (u )0

42( 12 )tan (u )cos (u )tan (u )

du

Reei.ir


5/23



42(1

2) 1

cos (u)du

%tili8ar la si*ie.te ie.tia/1

cos (x)=sec (x )

42(1

2) sec (u )du

%sar la i.teral ,o./ sec (u )du=ln ( tan (u )+sec (u ))

42(12 ) ln ( tan (u )+sec (u))

S*stit*ir/u=sec1( 12x)

42(1

2) ln( tan(sec1( 12x))+sec(sec1( 12x)))

Si3lii,a.o/

4 ln(1 4x2x2

+x

2)

xx24

dx=x24

xx24

dx

A3li,ar la s*stit*,i. e *.a i.teral/ f(g (x ))g'(x )dx= f(u)du 7 u=g (x )


6/23



u=x24

du=2

xdu

dx= 1

2xdu

xu

1

2xdu

12

u

du

Sa,ar la ,o.ta.te/ af(x )dx=a f(x )dx

1

2 1

udu

1

u=u

12

1

2u

1

2 du

A3li,ar la rela e la 3ote.,ia/ xadx=

xa1

a+1 7 o.ea1

1

2

u12+1

12 +1

S*stit*6e.o u=x24


7/23



1

2

(x24 )12+1

1

2

+1

Si3lii,a.o/

x24

4 ln(1 4x2x2

+x

2)+x24+C

Area.o la ,o.sta.te

4 ln(1 4x2x2

+x

2)+x24+C

A3li,ar los l:ites/ 2

5

4+x

x24dx=4 ln( 5+212 )+21=0

(f(x )) limx a+( f(x ))

xb

f(x )dx=F(b )F(a )=lim

a

b

(4 ln(1

4

x2x

2 +

x

2 )+x24 )=0x 2

lim


8/23



x2

(4 ln

(1

4

x2x

2

+x

2

)+x24

)lim

limx a

[ f(x )g (x )]=limxa

f(x ) limx a

g (x )

x 2+ (x24 )

x 2+

(4ln

(1

4

x2x

2 +x

2

))+ lim

lim

x2+(4ln(1 4x2x

2 +

x

2))=0

lim

x2+(4ln(1 4x2x

2 +

x

2))

lim

limx a

[ cf(x )]=c limx a

f(x )

x2+( ln(1 4x2x2 +x2 ))

4lim


9/23



%sar la ,o.ti.*ia e ln (x ) e. x=2

%sar la 3ro3iea e los l:ites/

(f(x )) limta+( f(x ))

tb


a

b

Cal,*lar

x2+ 1

4

x2x

2 +

x

2

lim

ln

x2+(1 4x2x

2 +

x

2)=1

lim

x2+(1 4x2x

2 +

x

2)

lim

Ree3la8a.o el valor/

1 4

(2)2(2 )

2 +

(2 )2

Si3lii,a.o


10/23



1

4 ln (1)

Si3lii,a.o

40

0

x 2+ (x24 )=0lim

x2+ (x24 )lim

Ree3la8a.o el valor/

(2)24

Si3lii,a.o

0

x5(4 ln(1 4x2x

2 +

x

2))=4 ln( 5+212 )+21

lim

x5(4 ln(1 4x2x

2 +

x

2)+x24)

lim


11/23



limx a

[ cf(x ) g (x )]=limx a

f(x ) limx a

g (x )

(4 ln( 14

x2x

2 +

x

2))+ lim

x5(x24 )

x5

lim

x 5

(4 ln

(1

4

x2x

2 +

x

2

))=4 ln

(5+21

2

)lim

x5(4 ln(1 4x2x

2 +

x

2))

lim

limx a

[ cf(x )]=climx a

f(x )

x5( ln(1 4x2x

2 +

x

2))

4 lim

%sar la 3ro3iea e los l:ites/


tb


a

b


12/23



Cal,*lar

x5

(1

4

x2x

2 +

x

2

)lim

ln

x5(1 4x2x

2 +

x

2)=5+21

2

lim

x5(1 4x2x

2 +

x

2)

lim

Ree3la8a.o ;/

1

4

(5)2

2 + (

5 )2

Si3lii,a.o

5+21

2

4 ln

(5+21

2

)

x5 (x24 )=21

lim


13/23



x 5 (x24 )lim

Ree3la8a.o ;/

(5 )24

Si3lii,a.o

21

4 ln(5

+21

2 )+21

4 ln( 5+212 )+210

Si3lii,a.o

21+4 ln(5+212 )


14/23



5ara resolver iere.tes ti3os e i.terales es i.is3e.sable te.er e. ,*e.ta las 3ro3ieaes

b

Eval*ar las si*ie.tes i.terales/

E'ERCICIO 2

sec2 (x )x

dx

Sol*,i./

sec2

(x )xdx=2tan (x )+C

sec2 (x )x

dx

A3li,ar i.teral 3or s*stit*,i./ f(g (x ))g'(x )dx= f(u)du 7 u=g (x )

u=x

du= 1

2xdx

dx=2udu

sec2 (u)u

2udu

2 sec2 (u ) du



15/23



2 sec2 (u ) du

A3li,ar la rela e i.tera,i./ sec2

(u )du=tan (u )

2 tan (u )

S*stit*ir e. la e,*a,i. u=x

2 tan (u )

Arear *.a ,o.sta.te a la sol*,i.

2 tan (u )+C


16/23



E'ERCICIO 4

1

4

1

(1+x )dx

Sol*,i./

1

4

1

(1+x )dx=2+ln (4 )ln (9)

1

4

1

(1+x )dx

Cal,*lar la i.teral i.ei.ia/ 1

1+xdx=2x2ln (x+1 )+C

11+x

dx

A3li,ar i.teral 3or s*stit*,i./ f(g (x ))g'(x )dx= f(u)du 7 u=g (x )

u=x

d u= 1

2x

du= 1

2udx

d x=2udu

11+x

2udx


17/23



2 2u+1

dx

A3li,ar la rela e la s*a/ f(x )g (x )dx= f(x )dx g (x )dx

2du 2u+1

dx

2du=2u

2du

&.teral e *.a ,o.sta.te/ f( a ) dx=xf( a)

2u

2u+1

dx=2ln (u+1 )

2u+1 dx


2 1u+1

du

A3li,ar i.teral 3or s*stit*,i./

f(g (x ))g'(x )dx= f(u )du

7

u=g (x )

v=u+1

dv=1du


18/23



du=1dv

21

v 1du

2 1v

du

A3li,ar la rela e i.tera,i./ 1

vdv=ln (v )

2 ln (v )

S*stit*ir e. la e,*a,i.v=u+1

2 ln (u+1 )

2u2 ln (u+1 )

S*stit*ir e. la e,*a,i. u=x

2x2ln (x+1)

Arear *.a ,o.sta.te a la sol*,i.

2x2 ln (x+1 )+C

A3li,ar los l:ites/ 1

4

1

1+xdx

1

4

1

1+xdx=4ln (9 )(2ln (4 ) )


19/23




tb


a

b

x1 (2x2 ln (x+1 ) )=2ln (4 )

lim

x1 (2x2 ln (x+1 ) )

lim

A3li,ar la rela e la s*a/ f(x )g (x )dx= f(x )dxg (x )dx

x 1 (2 ln (x+1) )

x 1 (2x )lim

lim

x 1 (2x )=2lim

x 1 (2x )

lim

Ree3la8ar la variable

2(1 )

Si3lii,a.o

2


20/23



limx 1

(2ln (x+1 ) )=ln (4 )

limx 1

(2ln (x+1 ) )

limx a

[ cf(x )]=climx a

f(x )

2 limx 1

( ln (x+1 ) )


5ro3iea e l:ites ,o.ti.*os/ Si f(x) es ,o.ti.*o alreeor e x=g(a) 6 g(x)

es ,o.ti.*o alreeor ex=a

7 e.to.,eslimx a

f(g (x ))=f (limx a g (x ))

Cal,*larln( limx 1 (x+1 ))

limx 1 (x+1 )=2

limx 1

(x+1 )

S*stit*ir variable

(1 )+1

Si3lii,a.o/

2

2 ln (2 )

Si3lii,a.o/


21/23



ln (4 )

2

ln

(4

)

limx 4

(2x2ln (x+1 ) )=4ln (9 )

limx4

(2x2ln (x+1 ) )

A3li,ar la rela e la s*a/ f(x )g (x )dx= f(x )dxg (x )dx

limx 4

(2x )limx 4

(2ln (x+1 ) )

limx 4

(2x )=4

S*stit*ir variable

2(4)

Si3lii,a.o

4

limx4

(2ln (x+1) )=ln (9 )

limx 4

(2ln (x+1 ) )

limx a

[ cf(x )]=climx a

f(x )

2 limx 4

( ln (x+1 ) )


22/23




5ro3iea e l:ites ,o.ti.*os/ Si f(x) es ,o.ti.*o alreeor e x=g(a) 6 g(x)

es ,o.ti.*o alreeor e x=a 7 e.to.,es

limx a

f(g (x ))=f (limx a g (x ))

Cal,*larln( limx 4 (x+1 ))

limx 4

(x+1 )=3

limx 4

(x+1 )

S*stit*ir la variable

(4 )+1

Si3lii,a.o

3

2 ln (3 )

2 ln (3 )

Si3lii,a.o

ln (9 )

4 ln (9 )

4 ln (9 )(2ln (4 ))

4 ln (9 )(2ln (4 ))


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Si3lii,a.o

2+ln (4 )ln (9 )

3 Aporte 199 Trabajo Fase 2

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