CALCULO INTEGRAL.doc

5/20/2018 CALCULO INTEGRAL.doc

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CALCULO INTEGRAL

OBJETIVO GENERAL,

CAPACITAR AL ALUMNO EN LA EJECUCION Y SELECCIN DEALGORITMOS PARA INTEGRAR FUNCIONES DE FORMA INMEDIATAAPLICANDO METODOS ADECUADOS E INTERPRETARGEOMETRICAMENTE LA INTEGRAL DEFINIDA, CALCULANDO LASMAGNITUDES QUE REPRESENTAN LAS VARIABLES INVOLUCRADAS ENSUS APLICACIONES; ASI COMO A LAS CONDICIONES DE CONTINUIDADY DIFERENCIABILIDAD DE LAS FUNCIONES.

TIEMPO ASIGNADO 75 HORAS.

TOTAL DE OBJETIVOS ESPECIFICOS 25.


2/74

OBJETIVO I.

EXPLICARA EL CONCEPTO DE DIFERENCIAL Y SU INTERPRETACIONGEOMETRICA SIN ERROR.

LA DIRENCIAL DE UNA FUNCION. SU NOTACION ES DF !" Y SE LEEDIRENCIAL DE F DE X O DIRERENCIAL DE LA FUNCION.

LA DIFERENCIAL DE UNA FUNCION ES EL PRODUCTO DE SUDERIVADO POR EL INCREMENTO DE LA VARIABLE INDEPENDIENTE,

df (x) = f(x) axx

xy

4

43 2 =

ax =dxdf(x) =f (x) dx

2

2

)4(

)4()43()4(

'x

xdxdxdx

dx

y

=

Y=6X2 +5Y= 12x

idemdx

dy =

XDXDY 12=

dy= (12x)(dx)

dy= dxxx )23)(424( 22

INTERPRETACION GEOMETRICA DE LA DIFERENCIAL.


3/74

R

Q

R

Ay

Ax

0 A B

OA VALOR INICIAL DE LA VARIABLE X

AT VALOR INICIAL DE LA FUNCION Y

AB # TP# INCREMENTO DE LA VARIABLE $! # A!

PR # PQ # QR# INCREMENTO DE LA FUNCION Y

SI $% & A%; LA CURVA ES CONCAVA HACIA ABAJO

SI $% ' A%; LA CURVA ES CONCAVA HACIA ARRIBA

OBJETIVO II


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SE OBTENDRA LA DIFERENCIAL DE DIFERENTES FUNCIONES.

LA DIFERENCIALYSE PUEDE HALLAR APLICANDO SU FORMULA DEDEFINICION O BIEN POR MEDIO DE LAS REGLAS DE CALCULOS DEDERIVADA.

ALGUNAS DE ESTAS SON(

dx

d.1 SEN u = CS u

dx

d(u)

2.dx

dCS u = !SEN u

dx

d(u)

3.dx

d"# u = SEC 2 u

dx

d(u)

4.dxd C"# u = !CSC 2 u

dxd (u)

5.dx

dSEC u = SEC u "# u

dx

d(u)

6.dx

dCSC u = !CSC u C"# u

dx

d(u)

E$E%&S


5/74

1.dx

dy= SEN 2 (3x 2)

dx

dy= SEN (3x 2) 2

dxdy

= 2 [ ])23( xSEN 12

dx

d

(3x 2)

dx

dy = 2 [ ])23( xSEN (3)

dx

dy= 6 SEN 3x 2

dy = (6 SEN 3x !2)(dx)

2.dx

dy= 9 SEC

3

x

dx

dy= *

dx

d(SEC

3

x)

dx

dy= *

)33

xTG

xSEC

dx

d3

x

dx

dy= * SEC

3

x"#

3

x(3

1)

dx

dy= 3 SEC

3

x"#

3

x

dy = (3 SEC 3x "# 3

x ) dx

3. y = 5 COS x

dx

dy= 5

dx

d(CS x)

dx

dy= 5

)(xdX

dSENx

dx

dy= 5 SEN x

dy = (5 SEN x) dx

4. Y = 7 SEN 52x


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dx

dy=

dx

d(SEN

5

2x)

dx

dy= (CS

5

2x

dx

d(

5

2x))

dx

dy= (CS

5

2x)

dx

dy=

5

14CS

5

2x

dx

dy= (

5

14CS

5

2x) dx

5. Y = TG 4

1

x

dx

dy=

dx

d("#

4

1x)

dx

dy= (SEC2

4

1x)

dx

d4

1x

dx

dy= (SEC2

4

1x)

4

1

dx

dy=

4

1SEC2

4

1x

dy = (SEC 2 4

1

x) dx

6. Y = xCOT2

dx

dy=

dx

d( xCOT2 )

dx

dy=

)2(

)2(2

XCOT

xdx

dxCSC

dx

dy

= XCOTdx

dxCSC

//

/

22

)2(2

dy = (XCOT

xCSC

/

2

2) dx

7. Y =4

3 2xCSC =

4

1CSC 3x2

dxdy

= 41

dx

d

(CSC 3x2

)


7/74

dx

dy=

4

1

)3(3x3xCSC 222 x

dx

dCTG

dx

dy=

4

1(! CSC 3x2 C"# 3x2 ) (6x)

dy = (! 46x

CSC 3x2

C"# 3x2

) dx

,. Y = SEN ax2

dx

dy=

dx

d(SEN ax2 )

dx

dy= (CS ax2

dx

d(ax2 ))

dx

dy= (CS ax2 ) (2ax)

dy = ( 2ax CS ax2

) dx

9. Y = tCOS2

dx

dy=

tCOS

tCOSdxd

22

)2(

dx

dy=

tCOS

tdx

dtSEN

22

)2(2

dx

dy=

tCOS

tSEN

22

22

//

dy = ( tCOStSEN

22 ) dx

10. Y = SEN x COS x

dx

dy=SEN x

dx

dCS x + CS x

dx

dSEN x

dx

dy= SEN x (!SEN x)

dx

d(x) + CS x (CS x)

dx

d(x)

dx

dy= SEN x (!SEN x +CS x (CS x))

dxdy

= (!SEN2

x) + (CS2

x) dy = xCOSxSEN 22 + dx

11. Y = 7x2 + 2x + 5

dx

dy= 14x + 2

dy = ( 14x + 2 ) dx

12. Y = xx +243


8/74

dx

dy= 22 )4(3

1,

xx

x

+

+

dy = ( 22 )4(3

1,

xx

x

+

+) dx

TAREA

1.- Y = = (CSC 20) 21

2.- Y = x 2 SEN x

3.- Y =x

COSx

4.- Y = 3 SEN 2X

5.- Y = 4 COS2

1x

6.- Y = 4 TG 5x

.- Y =4

1CTG !x

!.- Y = " SEC3

1x

".- Y =4

1CSC 4x

10.- Y = SEN x #x COS X $X 2 $4x $3

11.- Y = 0SEN

12.- Y = SEN 2 (3x-2)

13.- Y =2

1TG x SEN 2x

14.- Y = SEN 2x COS x15.- Y = TG 2x

TALLER

1.- Y = tCOS2 = (COS 2t) 12

dx

dy=

tCOS

tdx

dytSEN

22

)2(2


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dx

dy=

tCOS

tSEN

22

22

//

dy = (tCOS

tSEN

2

2) dx

2.- Y = tTG3 36 = (TG 36t13

dx

dy= 2

2

)36(3

3636

tTG

tSEC

dy = ( 22

)36(3

3636

tTG

tSEC) dx

3.! Y = x CS x

dx

dy= x

dx

dCS x + CS x

dx

dx

dx

dy= x (!SEN x

dx

d(x) + CS x

dx

dy= x (!SEN x) + CS x

dy = ( !x SEN x + CS x ) dx

4.- Y =SEC 3 x = (SEC x 13

dx

dy= SEC 2 x "# x

dx

d x

dx

dy= SEC 2 x "# x (

x2

1)

dx

dy=

2

1x SEC 2 x "# x

dy= (2

1x SEC 2 x "# x ) dx

5.- Y =4

1COS 20

dx

dy=

4

1

dx

d= (CS 20)

dx

dy=

4

1( !SEN 20

dx

d(20)

dx

dy=

4

1(!SEN 20)

dy = ( !41 SEN 20 ) dx


10/74

6.- Y =x

SENx

dx

dy=

2

)(

x

SENxdx

dxx

dX

dSENx

dx

dy=

2

)(

x

xdX

dxCOSxSENx

dx

dy= 2x

xCOSXSENx

dy = ( 2x

xSENxCOSX) dx

7.- Y = SEN 3x COS 2x

dx

dy= CS 3x

dx

d(3x) + (!SEN 2x

dx

d(2x))

dx

dy= 3CS 3x + (!2 SEN 2x)

dx

dy= 3CS 3x 2SEN 2x

dy = (!6 CS 3x SEN 2x ) dx

,.- Y = CTG (1 2X2

)

dx

dy= -CSC 2 ( 1 2X2 )

dx

d( 1 2X2 )

dx

dy= -CSC 2 (1 2X2 ) (!4x)

dx

dy= 4x CSC 2 (1 2X2 )

dy = (4x CSC 2 ( 1 2X2 ))dxOBJETIVO III

SE APLICARA LA DIFERENCIAL EN RESOLVER PROBLEMAS QUEINVOLUCRAN EL CALCULO DE VALORES APROXIMADOS.

CALCULOS APROXIMADOS DE INCREMENTO

).* SI $! # +!, EL INCREMENTO ES RELATIVAMENTE PEQUEO CONRESPECTO A ! EL VALOR DE +!, SE PUEDE OBTENERAPROXIMADAMENTE HALLANDO $%.E-/01(

1.! SEA Y = x2 + x + 1 y x S-RE -N /NCRE%EN" ESE x=2 AS"A x=2.01 ) y = x2 + x + 1


11/74

y = 12.01)(2.01 2 ++ y = [ ]12.014.041 ++ ! [ ]1)2((2)2 ++ y = .05 ! y = 0.05

) AAR E /NCRE%EN" &R%E/ E A /ERENC/A. Y = x2 + x +1 dy = (2 (2)+1) (0.01) = 2x +1 dy = (5) (0.01) dy = (2x +1 ) dx dy = 0.05

2.!SEA Y =2

1x2 + 3x Y S-RE -N /NCRE%EN" ESE x=2 AS"A x=2.,

) AAR A AR/AC/N REA CRRES&N/EN"E A dx .

Y =2

1x2 3x

y =21 (2.,)2 + 3(2.,)!

+ )2(3)2(21 2

y =2

1(.,4) + ,.4 !

+ 6)4(2

1

y = 3.*2 + ,.4 !2 !6 y = 4.32

) AAR E /NCRE%EN" &R%E/ E A /ERENC/A.

Y =2

1x2 + 3x

dxdy = 2

2 x + 3

dy = (x +3) dxdy = (2+3) (0.,)

dy = 4.4

3.! AAR A&RX/%AA%EN"E A AR/AC/N EX&ER/%EN"AA &R

E -%EN E -N C-B E AR/S"A x C-AN ES"E SE /NCRE%EN"A17.

= . . . dx = 0.01

Y = x 3

dxdy

= x3

dy = (3x3

) (0.01x)


12/74

dx

dy= 3x 3 dy = 0.05 x 3

dy = (3x 3 ) dx

TAREA

).* HALLAR CON LA AYUDA DEL CALCULO DIFERENCIAL ELINCREMENTO DE(

a) Y = x2 + 2x ! 6 x= 4 y dx = 0.,

8) Y =3 x2 + 10 x2 ! , x=2 y dx = 0.04

9) Y = 10 x2 + 6x + 10 x= 6 y dx = 2.6

d) Y = 3 x2 + 4 26x x= 2 y dx = 0.5

:) Y = 4 3 3x + * x= * y dy = 2.1

2.* UN DISCO METALICO SE DILATA POR LA ACCION DELCALOR DE MANERA QUE SU RADIO AUMENTA DE 5 A 5.34 CM.

a) HALLAR EL VALOR APROXIMADO DEL INCREMENTO DELA .

OBJETIVO IV

SE EXPLICARA EL CONCEPTO DE INTEGRAL Y LA CONSTANTE DEINTEGRACION.

CONSIDEREMOS LAS SIGUIENTES 6 CLASES DE EXPRESIONES(

Y = x4 Y;= 4x 3 dy = (4x 3 ) dxY = 2x Y;= 2 CS 2x dy = (2 CS 2x)

Y = :

xSEN4

Y;= 4 :

xSEN4

(CS 4x) dy = (4 :

xSEN4

CS 4x)


13/74

SE ADVIERTE CLARAMENTE QUE LA COLUMNA BY CSONRESPECTIVAMENTE LAS %ERIVA%AS Y %I&ERENCIALES DELAS FUNCIONES QUE FIGURAN EN LA COLUMNA A.

EN LOS CASOS QUE SE OBSERVAN DE LA COLUMNA ASE

CONSIDERA COMO LA FUNCION PRIMITIVA O LAANTIDERIVADA O BIEN LA FUNCION INTEGRAL LA CUAL SEEXPRESA.

+= Cxfdxxf )()(

LA INTEGRACION ES UNA OPERACIN CONTRARIA A LADERIVADA O A LA DIFERENCIACION.

LA C SE LLAMA CONSTANTE %E INTEGRACION Y ES

UNA CANTIDAD INDEPENDIENTE DE LA VARIABLE DEINTEGRACION YA QUE PODEMOS DAR A CCUANTOS VALORESQUERRAMOS; SABIENDO QUE SI UNA EXPRESIONDIFERENCIAL DADA TIENE UNA INTEGRAL Y TIENE TAMBIENUNA INFINIDAD DE INTEGRALES QUE DIFIEREN SOLO EN LASCONSTANTES.

OBJETIVO N. 5

S: d?@ >:?a: >d:f>da uDa>d< f


14/74

a >:?a9F> : u> G?< ::>9a:>: d: :>ay@:

1.! : a :xG?:F> G


15/74

TAREATAREA

a) (x(x 3!12)!12) 3dxdx8) (2x(2x 2!5x+3)dx!5x+3)dx

9) (x(x3

+5x+5x2

!4M4) dx!4M4) dxd) xdxxdx:) 1Mx1Mx2dxdxf) (4x(4x 3+3x+3x 2+4x!3)dx+4x!3)dx) (x(x 3!x!x2+1Mx+1Mx2!1Mx!1Mx 3)dx)dx) dxM2xdxM2x) dxM3xdxM3x

J) (x(x 3!6x+5)Mxdx!6x+5)MxdxO) x!xM2+2Mxdxx!xM2+2Mxdx) 4x4x3+3x+3x2+2x+5dx+2x+5dx) (3!2x!x(3!2x!x 4)dx)dx>) (x(x 2!1)!1)2dxdx


16/74

a+8xdx =a+8xdx = (a+8x)(a+8x)1 M 21 M 2

dxdx

u = a+8x 1M8u = a+8x 1M8 (a+8x)(a+8x)

1 M 21 M 2

8dx8dx

u = 8 1M8(uu = 8 1M8(u1 M 2 + 11 M 2 + 1

)M1M2+1 = (a+8x))M1M2+1 = (a+8x)3 M 23 M 2

M3M28 + 9M3M28 + 9

> = P 2(a+8x)> = P 2(a+8x)3 M 23 M 2

M38 + 9M38 + 9

(2+x(2+x22

)2dx)2dx

(4+4x(4+4x22

+2x+2x22

)dx)dx

44 dx+4dx+4 xx22

+2+2 xx22

dxdx

4x+4x4x+4x33

M3+2xM3+2x33

M3 + 9M3 + 9

u = 2+xu = 2+x22

1M2 1M2 (2+x(2+x22

))22

dxdx

u = 2x P(uu = 2x P(u2 + 12 + 1

)2+1 + 9)2+1 + 9 = (2+x22)33M6 + 9

"a?:a

a )a ) 6dM(5!36dM(5!3

22

))

22

8 )8 ) (x(x33

+2)+2)22

3x3x22

dxdx

9 )9 ) (x(x33

+2)+2)1 M 21 M 2

xx22

dxdx

d )d ) ,x,x22

dxM(xdxM(x33

+2)+2)33

: ): ) xx22

MMxMMx33

+2 dx+2 dx

f )f ) 3 1!2x3 1!2x22

dxdx


17/74

A01:@$ 1 ?9/>1 5A01:@$ 1 ?9/>1 5

duMu = > u + 9duMu = > u + 9

EJ:G (x + 2) + 9

u = x + 2u = x + 2

u = 1u = 1

xx22

dxM1!2xdxM1!2x33

1M6 1M6 6x6x22

dxM1!2xdxM1!2x33

u = 1!2xu = 1!2x

33

= >(1!2x = >(1!2x

33

)M6)M6u= !6xu= !6x

22

dxM3x+1 =1M3dxM3x+1 =1M3 3dxM3x+13dxM3x+1u = 3x + 1 =1M3 > u + 9u= 3 =>(3x+1)M3 + 9

T9a) dxM3x!2dxM3x!2

8) dxMxdxMx

22

+,+,9) xdxM4xxdxM4x22

!3!3


18/74

d) dxM5x+1dxM5x+1:) xdxM4xxdxM4x

22

+3+3f) xx

22

+x+1dxMx+2+x+1dxMx+2) xx

22

!5x+6Mx!3!5x+6Mx!3

F9/>18 (:9 u + u)+9:9 udu = >(:9 u + u)+9. :9:9

22

udu = u + 9udu = u + 9#. 9999

22

u du = !9


19/74

1.!1.! :> 3xdx:> 3xdx

u = 3xu = 3x

u = 3u = 3

= 1M3= 1M3 :> (3x)3dx:> (3x)3dx

= 1M3(!9(ax+8)dx

u = ax + 8u = ax + 8

u = au = a

5Ma(:> (ax+ 8)a)dx !59 (ax+ 8)a)dx !59 :9 u) + 9P(> :9 u) + 9

> :9(x> :9(x22

+1)M2 + 9+1)M2 + 9

5.!5.! 9 xM2dx9 xM2dx


20/74

u = xM2u = P

2 9 xM2 (1M2)dx9 xM2 (1M2)dx

2 (> :> u)+ 92 (> :> u)+ 92 > :> xM2 + 9

T9

:>(4x:>(4x22

+3)xdx+3)xdx

:>(ax+9)dx:>(ax+9)dx

:> xM4dx:> xM4dx

9


21/74

A1=>8 :$


22/74

13. :9 =

14. 1 + 2

= :92

15. 2

= :92

!1

16. =

1.992 =1+92

1,.99 =

1*.92 +1 = 992

20.92 =992 !1

21.9 =

22.:>(x+y) = :> x 9 x 9


23/74

-


24/74

09 :8 $:?9@:18


25/74

u = :>5x 1M5(u>+1M>+1) + 9u = 5945xM4 + 9

:>45xM20 + 9

T!E

xdx39


26/74

D$ 18 198 $ / % , $ $ 89 /8 /98 0986D dD = (S:>2D dD)2S:>2D dD

a8:2V = P ! P C


27/74

(1M2) (!3M,) C 2D + C

"AREA

1.! C4DdD

3.! S:>2x C2xM2 C


28/74

9< u 9< >u du d: > y a8:> >9< u 9< >u du d: > y a8:> > 9< u :>< >u du9< u :>< >u du


29/74

8)8) u. ? u =! P 9) I +1M29) I:> u :> u =! P 9) I +1M29) I

[!1M29(4!3)D\dD[!1M29(4!3)D\dD

!1M29


30/74

1.- T 3 x dx

" 2 x " x dx

S"E/OS

Tg 2 x = Sec2 x -1

(S:9 2 x!1) " x dx

" x S:9 x dx ! " x dx

= " 11 x ! S:9 x + 9 1 + 1

" 2 x ! S:9 x + 9 2

2.- T 4 2x dx

( " 2 2x " 2 2x dx)

S"E/OS

Tg 2 2x = Sec 2 2x -1

( S:9 2 2x !1) " 2 2x dx A B

" 2 2x S:9 2 2x ! " 2 2x dx

- = 2xu = 2


31/74

A 1 " 2 2x S:9 2 2x (2) dx

2

" 12 2x = P " 3 2x = " 3 2x + C 2+1 3 6

B " 2 2x dx = P " 2 2x (2) dx

- = 2x P " 2x ! 2x + C u = 2

" 2x ! 2x + C 2

3.- Ct 3 X dx = -3 Ct 2 x - -3 * S' X + C 3 2 3 3

C 2 X C X dx3 34

S"E/OS

Ct 2 x = C, 2 x - 13 3

( C9 X 1) C X dx 3 3

C X C9 2 X dx ! C X dx 3 3 3

- = X - = X 3 3

du = 1M3 du = 1M3


32/74

3 C X C9 2 X (1M2) dx 3 C X (1M3) dx 3 3 3

!3 C X + C = S:9 u + 9

2 3 = ! 3 S:> x + 9 3

4.- Ct 3 2x dx

C 2 2X C 2x = C 2x C9 2 2x dx ! C 2X dx

- = 2x - = 2x u = 2 du = 2

S"E/OS

Ct 2 2x = -C, 2 2x 1

= P C 2x C9 2 2x (2) dx = P C 2x (2) dx

= ! P C 2 2x + C = P S:> 2x + C 2

! C2

2x + C 4

5.- Ct 3 2x C, 2x dx =! C9 3 2x ! C 2x + C 6 4

CC 2 2x C 2x C9 2x dx =2x C 2x C9 2x dx = (C9(C9 2 2x 1) C 2x2x 1) C 2xC9 2xC9 2x

S"E/OS = C9 2 2x C 2x C9 2x - C 2x C9 2x


33/74

CtCt2

2x = C,2x = C,2

2x 12x 1 - = 2x- = 2x

- = 2x- = 2x

- = 2 u = 2

= P C9 2x = P C 2x 2+1 1+1

= ! C9 3 2x + C = C 2 2x +C 6 4

6.

dx = " 2 x dxC x

S"E/OS

Ct 2 x = T2 x dx

TAREA

1 .! C9 3 x C xdx

2 .! "2 x S:9 2 x dx

3 .! 15 "5 (2x) S:9 2 (2x) dx

4 .! 13 C4 (,x!1) C9 2 (,x 1 ) dx

5 .! " 4 x S:9 2 x dx

6 .! " x S:9 3 x dx 2 2

.! C3

2x dx


34/74

,.! " + x dx

*.! C 5 2x dx

10.! (1+" )3

11.! C 6 3 d

OBJETIVO 13 Y 14

/>:?a?a 9


35/74

= 2 " x + C 2 " 3 + C2 3

= 2 " x + C

2

2 .- S' 6 ax dx

( S:9 2 ax ) 2 S:9 2 ax dx

S"E/OS

S' 2 ax = 1 +T2 ax

( 1+ " 2 ax ) 2 S:9 2 ax dx

( 1+2" 2 ax + " 4 ax) S:9 2 ax dx

S:9 2 ax dx + 2 2 ax S:9 ax dx + " 4 ax S:9 2 ax dx

- = ax - = ax - = ax

u = a - = a - = a

1Ma S:92 ax (a) dx + 2Ma " 2 ax S:9 2 ax (a) dx + 1Ma "4 ax S:92 ax (a)

dx

1Ma " ax + C 2Ma " 3 ax + C 1Ma " 5 ax +C

3

5 " ax + C 2 " 3 ax + C " 5 ax + C


36/74

a 3a 5a

3.- C, 4 2 d

C


37/74

S' S'2

S"E/OS

1 = C,2

S'2

5.- C, 4 5x - 3 dx= !2M5 C 5x 3 + 2 C 3 + C2 15

C9 2 5x 3 C9 2 5x 3 dx

2 2S"E/OS

C, 2 5x - 3 = 1 + Ct2 5x - 32 2

( 1 + C2 5x ! 3 ) C9 2 5x 3 dx 2 2

A B

C92

5x ! 3 dx + C2

5x ! 3 C92

5x ! 3 dx 2 2 2

- = 5X ! 3 - = 5X ! 32 2

- = 5x ! 3 - = 5x ! 3 2 2 2 2

du = 5M2 du = 5M2

du = 2M5 du = 2M5

2M5 C92 5x ! 3 (5M2) dx2

= 2M5 C 5x ! 3 2


38/74

"

= C2 5x ! 3 C9 2 5x ! 3 dx2 2

= 2M5 C2 5x ! 3 C92 5x ! 3 (5M2) dx2 2

= 2M5 ( C 3 5x 3 ) + C 3

= 2 C 3 5x !3 + C 15

T!E


39/74

1.! S:9 6 x dx

2. .! C9 6 D dD 2

3 .! S:94 y dy 2

4 .! S:94 2 d

5 .! C9 2 3x dx 2

6 .! C9 2 5x dx

.! S:92 5x ! 3 dx 2

, .! 4 C9 4 d 4

* .! 5 S:9 6 3x 2 dx

10.! 3 C9 6 2x ! 5 dx 4

/TOO E $NTEG!C$N

B$E"/ 15

/>:?a fu>9: >d:f>da ?:du98: a >:daa G


40/74

:?a: Hu: 9:>:> :xG?:: d: cbxax ++2 < bxax +2

Gu:d:> >:?a?: :.

1:? %Ka >:?a G9a u>a :xG?:F> d: 2d< ?ad< d: 3

K?>a:xG?:F> d: d a:8?a9a).

EJ:G


41/74

[ ]

.2

4

42

.442)2(2

142

2

2!x2

2)2(

2)1(2)2()2()2(2

1)2(2)2()2(

2

12)2(2

42)2(

.)22()2(2

122

2

41x4)442(

(2)2.!2)2!(x42.4

)22(2122

2122

152xx2!1x:>2a1

4a1)2(xu

4a22)1(2

42)1(

15)122(

(2)21)2(x522.3

1!x

2!x

:>1M2a2M3

1M4a3M1)2!9xu2

1M2*2)2M34(2

4

12)

2

3(

==

=

++++==

=

++==

++++

=+

+++++=+

++++=+=

=+=

=+=++

+++

++++

==

=

==

a

a

a

cxxxenxxxxu

xu

xxenzxxxu

x

cauuenauxx

cx

cauuenaauuduau

f"!mu#a

cxu

xu

x

xx

xx

tcxu

u

x


42/74

[ ]

[

[ ]

[ ]

[ ]

3

*

*2

2

2)2( 2)2(2 *2*2( 442 442 .542 .542 )1(

542

45

3.6

1

2)1(

2)1(2

24

42

2)1(4

223

223

223

23.5

=

=

=

+=

+= += + ++ ++ + +

+

+=

+=

+=

==

=

+

++

+

a

a

a

xu

x$ xux xx xx xx xx

xx

x

dx

Yu

yu

Yu

a

a

Y

YY

YY

yy

y

d!


43/74

[ ]

( )[ ]

3M4a16*Ma16

*

24

3

=2

*M16a24M316

*

16

*

2

322

.4M3

4M3

4M3

1

4

13M4)2(=u2)

2

32(=2

(!1)1.1

4

1

*M163M4)2(=

d=4)322(

2)4M3(42M3(=d=4

16

*2)

4

34[(=

d=2=2!3=

dx!,.

9.1M2 x.4

11

1a121M2x4

1a94M422

1

.1

)2M1(BA

1

1

4

12M1\4M14M5

4

124

.12

1

4

11M2)2(X -2)2M1(u2)4M52(4

).4

5

4

4xa(x2

224

1

12)2M1(4

15424

)121M2(x4

dx

5*+4.+

2

==

+

=+=

++

+=+=+

+

=++

=++

++=

++=

++=

=++

=

++

+

+=+

++

++=+=++

+++

=++

++

++++

tuCt

tca!ct

cua

ta!ca

t

ta!ca

cxa

tcx

a!cxuxx

ca

utcaxxx

au

dx

x

dxxx

yx

dx


44/74

2d< %K:?ad< : u>a f?a99F> 9uy< :>u:?ad


45/74

( )

++

+=

+++=++

+++++=++

++=+

+==

+===

==

==

==

=

+=

=

2541

)32(.5

.*

1

a2!u2u

du!4.

24

)(.4

2222

122

2

1a2!u4!3.

12

)1(.3

)22(22

122

2

1a2u2!2.

.2254

)310.2

.*22

122

2

1u9du!a2!1.

25),x!(x2

1)dx!(x!1.

>/>=:A?a9Fd:

.2162

3!2

.4

y!2a?A:>x.u4

x2u16a

x2u2162

.2

3!216

216

2

2

3

2163

2162

216

3

..22

2dxx2!162!

216

)23(2

xx

dxx

ca

usena!c

x

dxxs

cauuenaauuduxx

dxx

cauuenaauudux

dxx

ca

usena!cua

u

%"!mu#as

cxenxu

ca

a

enutcx&

x

xdx

x

xdx

x

dx

x

xdx

tca

usena!c

ua

dx

x

dxx


46/74

8J:I< 16.

/>:?a?a fu>9: >d:f>da ?:du98: a >:daa aG9a>d< : K ? f:??


47/74

..9


48/74

===

==

==

==

++=

+=

+=

++=+

+=+

=+=

+=+==

==

======

58

6a258

36a2252

6:95

6

5

6

:9*

10

* x

36225.4

9.x:>10

1

2292

1

2

23

42

4

1

44x2

x

CaE>


49/74

ctxxtxb

adxx

b

ax(aciend"

b

ab

ab

x

dx

:9:9

5

625

362252

36225.4

==

===

==

cx

xn

cxx

n

cxtxenxdx

xtxdx

txxtxdxxx

+

+

+

==

36

362255

5

1

36

36225

6

5

30

6

9aE8d


50/74

B$E"/ 1 1, y 1*

R::?a: >d:f>da G


51/74

EJ:G:?a

1) xsen&x

dxdu

xu

==

cxu

senx dxu

dxsenxdu

senxdu

+=

=

=

=

9

)(

2

22

22

2

22

22

2) xdx5B>


52/74

x

dxdu

xu

5

5B>

=

=

cx&

dx&

xdd&

dxd&

+=

=

=

=

cxx

cxxx

dxxx

x

dxxxx

&duu&&ud

+=

+=

=

=

=

)5

15(B>

5

15B>

5

15B>

5))(5(B>

)(

3) xdxx 3B>2

x

dxdu

xu

3

3B>

=

=

2

3

2

2

2

x&

dxx&

dxxd&

dxxd&

=

=

=

=

cxx

cx

xx

cx

xx

dxxxx

x

dxx

x

x

&duu&&ud

+

=

+

=

+=

=

=

=

*

13B>

3

2+3B>

3

3*

1)3(B>

3

*

1)3(B>

3

33)3(B>3

)(

3

33

33

33

33


53/74

CAS ///

dxxe x

dxdu

xu

==

ce&

dxe&

dxed&dxed&

x

x

x

x

+=

=

==

cxe

cexe

dxeex

&duu&&ud

x

xx

xx

+=

+=

==

)1(

)(

)(

dxex x

23

2

3

3xdu

xu

=

=

ce0

dxe&

dxe&

dxed&

dxed&

x

x

x

x

x

+=

=

=

=

=

2

2

2

2

2

2

1

)2(2

1

( )

dxex

dxexex

dxxee

x

&duu&&ud

x

xx

xx

22

2223

222

3

2

3

2

3

2

322

)(

=

=

=

=

xdu

xu

=

= 2

ce

&

dxedx

dxed&

x

x

x

+=

=

=

2

2

2

2


54/74

( )

dxxe

dxxeex

dxxee

x

x

xx

xx

=

=

2

222

222

3

2

2

222

( )

ce

dxe

dxexe

dxee

x

x

x

xx

xx

+=

=

=

2

2

22

22

4

1

22

1

2

12

1

2

22

dxdu

xu

==

2

2

2

2

x

x

x

e&

dxed&

dxed&

=

=

=

cxxxe

cexeexex

x

xxxx

+

=

+

2

333

2

4

1

222

3

2

232

222223


55/74

CAS /

dxxx +1

dxdu

xu

==

( )( )

( )

( )2M3

1

12M1

11

1

1

2M3

2M1

2M1

2M1

x&

x&

dxxd&

dxxd&

+=

+++=

+=+=

( )( ) ( )

( )( )

( ) ( )c

xc

x

dxxxx

dxx

xx

&duu&&ud

+

+

+=+

+=

++

=

++=

=

+

12M3

1

3

2

3

12

13

2

3

12

3

1212

)(

12M32M3

2M32M3

2M32M3

( ) ( )

( )( ) cx

xx

cxxx

+++

+

+

+

2M52M3

2M52M3

115

4

3

12

2M5

1

3

2

3

12


56/74

1) dxa!csenx

21 x

dxdu

a!csenxu

=

=

cx&dx&

dxd&

dxd&

+==

=

=

( )

cxa!csenxxx

xdxxa!csenx

&duu&&ud

++=

=

=

9


57/74

3) dxxx 12

xdu

xu

2

2

== ( )

( )

( )

d&x

&

dxx

&

dxxd&

dxxd&

3

)1(2

2M3

1

1

1

2M3

2M3

2M1

2M1

=

=

=

=

( ) ( )

( )( )

( )

=

=

=

=

2M3

2M32M32

2M32M32

13

4

13

1

3

123

212

3

)1(2

)(

xx

xxxx

xxxx

&duu&&ud

1==

du

xu ( )

( )

( )

( )5

12

2M5

1

1

1

2M5

2M3

2M3

2M3

x&

x&

xd&

xd&

=

=

=

=

( ) ( )

( )( )

( )

( ) ( )

( ) ( ) ( )105

116

15

1,

3

12

105

116

+

12

15

,

12M5

1

15

,1

5

2

15

12

5

12

5

12

3

4

2M+2M52M32

2M+2M+

12M52M5

2M5

2M52M5

xxxxx

xc

x

cx

xxx

dxxxx

+=

+=+

=

+

+

=

=

=

+


58/74

R%-AS E RE-CC/N

a f


59/74

) +

+

++

= uduusenm

n

nm

uusenduuusen nm

mmnm 2

11

9


60/74

EJ:G


61/74

2)( ) + 2M521 x

du

2M3

1

2M5

1

===

=

==

m

xu

a

m

xu

a

( ) ( )( ) ( )

+

+

+

=

+

12222222

22

32

22

1mm

ua

du

m

m

uam

u

aua

du

( )( ) ( )( )

( ) ( )

( ) ( )

( ) ( )( ) ( )( )( ) ( )

( ) ( )

( )

( ) ( ) c

x

x

x

x

x

x

x

dx

x

x

x

dx

x

x

x

dx

x

dx

x

x

x

dx

x

x

++

++

=

++

=

++

+=

+

++

=+

=

++

+=

+

+

+=

2M122M32

2M12

2M122M12

12M3212M322M32

2M322M32

12M5212M52

13

2

13

013

2

11

0

13

2

122M32

32M32

122M323

2

13

2

13

2

13

122M52

32M52

122M521

1


62/74

3) ( ) + d&x2M32*

2M3

3

===

m

xu

a

( ) ( ) ( )

+

+

= d&au

m

ma

m

auuduau

m

mm 122

22222

12

2

12

( ) ( )( )( ) ( ) ( )( ) ( ) ( )( )( )

( ) ( ) dxxxx

dxxxx

2M12

2M32

12M322

22M322

*4

2+

4

*

312M32

32M32

12M32

3

+++

=

++

++

+=

1

3

=

=

b

a

( )( )

( )

( )[ ]cxttxxcxttxx

xxtx

xtx

xx

xxdxxtx

xaxb

adxxt

b

ax

dxx

+++=

+++=

+=

+=

=

==

==

+

3

3

2

2

2

2

2

2

3C:9B>*4

2+

3C:9B>*

C:9*C:9*

1C:9*

C:93C:93

C:93C:933

C:9C:9

*4

2+


63/74

4) dxx 39


64/74

6) xdxxsen 42 9


65/74

&a?a Ga? a 9a>: :> u>a >:?a : >:9:a?< 99F> = df2. a


66/74

2224

224

1213

23

231

2424

24

)(

byby

cyby

ydybdyy

dyyby

ybyy

++

+

=

++

2

2

2

224

224

24

024

02

4

4

16

02

)2(

4

)2(

2024

bc

cb

cb

cb

ycyby

+=

=+=+

=+

==+

/.! y1=6x3+ *x2+ x , If = y = !3

3

+,23

3

23

,23

*

46

,*6

,*6(

234

234

3

23

=

=+++

+++

++

++

x

cxxx

x

cyxxx

dxxdyydydxx

xxx

62

245.4,15.121+

+245.4,15.121

+242

*)2+(3

2

),1(3

+,2

)3()3(3

2

4)3(3 23

=+=

=+++

=++++

=+++

c

c

c

c

c

62,2

32

32

34 ++= xx

xxy

.! y1 = 4 ! ,x3 ! 6x4 ! x5 If = 20x = 4


67/74

4

2065

624

65

6

4

,4

6,4

)6,4(

654

654

543

543

=

=+

+

x

de!i&ada#acxxxx

cxxx

x

dxxdxxdxxdx

xxx

46.242+

66.6,2122,5121620

2066.6,2122,51216

206

)40*6(

5

)1024(6)256(216

206

)4()4(6)4(2)4(4

654

=+++=

=+

=+

=+

c

c

c

c

c

5431 6,4 xxxy =

a u:>: :xG?:: a:8?a9a:>: : :> d:?Ia>d< a u:>:fu>9:.

::?>a? a 9a>: d: a >:?a9F>. C

';:ada

d' *a

%a*; d' *a

%a;a8*' ?x@

%a*; d' *a

A> ?:


68/74

x

H

B

XGAXE

X

O

AD

I J

F

Y

a

a :9ua9F> d: a 9u?Ia AB y = _(x)

C 8a: a a fu?a y :a 9d a


69/74

B

XKEO

A

D

L

FY

a

b

C

La Inte$ral %efinida

: :


70/74

a df:?:>9a d: Ia


71/74

2. R::GaDa? a Ia?a8: d: a >:?a G


72/74

+=

+

4

)5(

3

)5(

2

)5()5(4

4324

4325

1

432xxx

x

( )[ ]25.03.05.04(25.15666.415.12204

)1(

3

)1(

2

)1()1(4

432

++=

+

[ ] 264.11,)45.30*.122( &==

f).!

=

=

3)4(

3

)2(

3

332

4

32

4

2 xdxx

224)33.2166.2( &=+=

1.! +=e

icxn

x

dx

[ ] 101 === inenxn ei

= e

i x

dx

1

2.! 2

0

C"s _ d_ = [ ] cSen +20

_

0e*002e1,0

2SenSenSenSenSen ==

=== 2

01d__101

C"s

3.! =404 d__

Sen

_=A1_ 22 +=Sen


73/74

+4

0

22 d___)S:>1(

t

+4 4

0 0

222 __S:>_d__

dtSen

4

0

3

_

3

=A_

+t

+

+

3

00

3

33 4

4t

tt

t

+

+

3

)0(0

3

33 4e1,0

4e1,0 t

tt

t

03

e45e45

3

+t

t

34

311 =+=

4.! 1

1

32)2( dxxx

dxxx )2( 32

++ dxxdxx 13122

1

1

43

432

xx

4)1(

3)1(2

4)1(

3)1(2

4343

)25.066.0()25.066.0( ++=*1.041.0 +232.1 &=

5.! dxxx

1

3 32

11

dxxx

32

11 R:9G?


74/74

+

=

+

2

21

)1(2

1

1

1)3(

2

1)3(

+

+=

+

1,

1

3

1

2

11

)3(2

1

3

)1(2

= 1.5 (!0.33+0.05)= 1.5 (!0.2,) = 1.5 + 0.2, = 1.,.

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CALCULO INTEGRAL.doc