Calculo Integral 02 - Bernardo Acevedo Frias

8/22/2019 Calculo Integral 02 - Bernardo Acevedo Frias

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f(x)dx | f (x) | d;a

De mo s t r a c i n .

Sea sab e qu e - | f ( x ) | < f ( x ) i | f ( x ) I , luego f ( x ) | d xa

|f(x)|dx ; y de aqui se con cl uy e lo que se que r a

1.13.11. Si f es cont inua en [- a, a] ent onc esa a

a ) f(x)dx f(x)dx, si f(x) es par en [- a, a]0

b) f( x) dx = 0,, si f(x) es un a fu nc i n im pa r-a

D e m o s t r a c i n ( e je r ci ci o )1

E j e mp l o . 1 La integral dx =0 ya que f(x) :( l + x a ) 4

un a f un c i n .i. m pa r en [-1,1 ]T T

Ejemplo .2 Dem ost rar que (x+ ISen xI )d; Senxd:"IT 0

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lucin.

Tr. TT TI( x + | S en x ) d x ' dx + J Se nx j dx = 0+ j S en x| d x

re -T C

ti en; - | d x Se nx dx .0

1. 13. 12 S ea f i nt e gr a bl e y pe r i di c a c on p er i o do t , e nt o nc e sb+t

f ( x ) d ;a+t

f ( x ) d x .a

[ ) ei nost r a e i n .

b+tf ( x ) d ;a+t

f ( x +1) d x f ( x | d ;a

Ej e mpl o . 1 Ha l l a r

So l uc i n.

1 s i 0 < X


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0 0d x + f ( x ) d x = 1 + - dx = 1- 1 = 0,

8Ej empl o - 2 Hal l ar f ( x) dx s i f ( x) = x 0< x


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3. Ver i f i c ar que ;.,3 | + ; , 3 ) D ; x 3 dx .0

4. Si f ( x ) d x0

f ( x ) d g ( x ) d x = - 1 g ( x ) d x = 40 0

Hal l ar

a ) f ( x ) d x b )01

d) 3 f ( x ) d x .

0

f ) f ( x ) d x +

( f ( x ) +2g( x ) ) dx . c )0

e ) j ( 2f ( x ) ~3g( x ) ) d:0

g ( x ) d x .

g ( x ) d :

5. Demos t r a r que 1 i 50

6. En l os s i g ui e nt e s e j e r c i c i o s c ont e s t a r con f a l s o o v er d ade rj us t i f i c ando s u r e s pues t a .

a ) s i f ( x ) d x > 0 e n t o n c e s f ( x ) > 0 p ar a t o da x e n [ a b ]

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b) s i f e s a c o t a d a e n [ a , ti ] e nt o nc e s f e s i n t e gr a bl e en [ a , b ] .| ) Si f e s i n t e gr a bl e en [ a, b J , f e s a c o t a da en [ a , b ] .d) s i f no e s c o nt i n ua en [ a , b ] , e n t o nc e s f n o e s i n t e gr a bl e en[ a , b ] .7. Cua l es de l as s i g ui e nt e s f u n c i o ne s s on i n t e g r a b l e s

.1a) f ( x ) =2 +S e nx +Co s x . b ) f ( x ) en [ 5 , 1 0] +3i Send. / ; ; ) s i x =4= 0c ) f ( K) = en [ >2, 2 ] . d ) f ( x ) = en

X .1 0 si . x =018. Dado que 2 i l - x 2 ) 1 / 2 d x = Tt h al l a r el v a l o r d e

[ - 2 , 2 ]

a) ( 9 - x a ) x / 2 d x b) ( l - ( x 2 ) / 4 ) 1 / 2 d x c ) ( 3 6 - 4 x ) x ' a d x0 0

9. S u po n er q ue h, f s o n c o n t i n u as y q ue f ( x ) d x = - 19

f ( x ) d x = 5; h ( x ) d x = 4; ha l l a r7

a ) [ 2f ( x ) - h( x ) ]dx-. b) [ f ( x ) +h ( x ) ] d x .71 . 1 5 T E O R E MA S F U N DA ME N T A L E S DE L C A L C U L O.Se e nc u e nt r a a ho r a en c o n d i c i o n es de e s t a b l e c e r 1 a r e l a c i n

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bs i c a q ue e x i s t e e nt r e i ne gr a c i n y d i f e nc i a c i n.Par a d es c r i b i r e s t a r e l a c i n, s e c o mi e nz a c on u na f u nc i n f q uees c o nt i n ua en el i nt e r v a l o c e r r a do t a , b ] . Pa r a c ada x de? [ a , b ] ,

l a i nt egr a l f ( t . ) dt e s un n me r o y en c o n s e c u e n c i a s e p u ed e

def i ni r u na f u nc i n F en [ a , b ] co mo F ( x ) : f ( t ) dt y as s ea

f or mmul a E l p r i me r t e o r e ma f u n d a me n t a l d e l c l c u l o q uedi c e as i : s i f es c ont i nua en [ a , b] , l a f unc i n F de f i n i d a en[ a, b]med i ant e F( f ( t ) d t e s c ont i nua en [ a , b] , di f e r e nc i a bl e en

( a, b) y s a t i s f a c e q ue F" ( x) = f ( x ) pa r a t o do x d e ( a , b)De mo s t r a c i nSe ve? un e s que ma g r f i c o de l a de mo s t r a c i n e n el c a s o e n quef >0; per o f puede s e r mayor o menor que 0 en [ a , b ] .

F(x+K) - ^

I I Ii ,a x+hfl h

D X

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F( x+h) = r ea de a a x + hF ( x ) = r ea de a a xF (x + h ) --F ( x ) - r e a d e x a x + hF ( x +h ) - F ( x ) r e a d e x a x +h

h f ( x ) si . h e s p e q u e o ; h >0,h

Si a. i x( x + h ) ~F ( x )x + h

fa

x + hf =

Se hace Mh = v a l o r m x i mo d e f en [ x , x +h ] = S u p Cf ( x ) / c 0 - h - >0H f ( x ) *hDe mo do s e me j a n t e s e pue de var . i f . i c ar que s i x *=( a, b] , h0~ h5 9


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F ( x +h) - F ( x )Si x G( a , b ) * y * * s o n v a l i d as y por l o t a nt o 1 i mh- >0 hf ( x) ; l uego F e s d i f e r e nc i a bl e en ( a, b) y por l o t ant o c o nt i nu aen [ a , b ] . De mo s t r e mo s q u e F e s c o n t i n u a e n [ a, b ] .

F ( a +h ) F ( a )De * l i m = f ( a) y e s t o i mpl i c a q ue l i m F ( a +h ) ~F ( a ) =0 yh- >0"- h h- >0-as i l i m F ( a +h ) = F ( a ) . E s t o d e mu e s t r a q ue F e s c o n t i n u a a l ah - >0 -der e c ha de a ; de f o r ma a n l o ga s i x =b e n s e t i e ne quel i m F ( b +h ) = F ( b ) ; y as i F e s c o nt i n ua en [ a , b] .h- >0~E1 t e or e ma a nt e r i o r s e pu ed e g en er a l i z a r a s i :s i f es c o n t i n ua en [ a , b ] y c e s un p un t o c u a l q ui e r a de Ca , b ] ; s e

v e r i f i c a q u e F" ( x ) f ( t ) dt es c ont i nua en Ca , b] , d i f e r e nc i a dl ecen ( a , b ) y F ' ( > ) =f ( x ) p ar a t o do x en ( a , b )

E j e mp l o . 1 S e a F ( x ) = t 3dt . ha l l a r F ' ( x )So l uc i n.F ' ( x ) =x 3 y a q u e f ( t . ) =t 3 e s c o n t i n ua en t o da s p ar t e s , enpa r t i c ul a r en u n i n t e r v al o c e r r a do q u e c o nt e ng a al i n t e r v a l oC- 2, x ]E j e mp l o . 2 S e a F ( x ) ; ( t 2 +l ) 1 / 2 d t s ha l l a r F ' ( x )

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Solucin.

F ' ( ) - ( x a +l ) * .

E j e mp l o . 3 Se a A(;;) = 1+Sent d t ; hai l ar A' ( x ) .0Q ( X )

So l uc i n. A( x ) 1+Sent dt = 1+Sent dt = F ( g ( x ) ) , l uego0 0A' ( x ) = F' ( g ( x ) ) . g' ( x ) = f ( g ( x ) ) . g' ( x ) - . g' ( x )1 +S en ( g ( x ) )

1 1. 2x ; s i e nd o f ( t. ) =l +Senx52 1+Sent

E j e mp l o . 4 S e a A ( x ) dt1 + t*

ha 11ar A' ( x )

So l uc i n.

f ( t ) = ; F ( x ) =1+t *

f ( t ) dt y as i A ( x ) =[ F ( x ) ] 3 l u eg o A ' ( )

3 [ F ( x ) ] = . F' ( x) = 3 [ F ( x ) ] a . f ( x ) f ( t ) d t - f ( x )

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E j empl o . 5 Sea A ( x )Sol uc i n,

f ( t. ) d tf ( t ) d t ha l l a r A ' ( x ) s i f ( t . ) =

1 + 1 =

F( x) = f ( t ) dt as i que A( x ) = F ( F ( g ( x ) ) ) =F ( F ( Xa ) ) ; l uego1

A' ( x ) =F ' ( F ( g ( x ) ) ) . F ' ( ( g ( x ) ) . g ' ( x ) =f ( F


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SenxSea A ( x ) ( u a +Co s u) d u, v er i f i c a r que

0A' ( x ) =( Se n2 x +Co s ( S en x) ) . Co s x

b3. S e a A( x ) l +u 2 +Senu du, v e r i f i qu e que A' ( x )

:

idu

l +u2 +Senu

4. Sea A ( x ) = Sen(y

Sen (0 0

5 e n 3 t . d t ) d y) v e r i f i que que A' ( x ) :

yC05( Sen ( Se n 3 d t ) dy ) . Se n( Se n 3 t d t ) .

0 0 0

5. En c a da c a s o c a l c u l a r f ( 2! ) s i f es c o nt i n ua y s a t i s f a c e l af r mu l a pa r a t o do x >0.

a) f ( t ) d t = x 2 ( 1 + x ) . b)0

f ( x )O t 2 d t = x 2 ( l +x )

0

f ( t ) d t = x 2 ( l +x ) .0

x 2 ( l +x )d ) f ( t ) d t

0

6. E nc o n t r a r u na f u nc i n f y un v al o r de l a c o n s t a n t e c t al q ue6 3


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t f ( t ) d t = Se nx - x Co s x - ( : : a/ 2 ) .) f ( t ) d t = ( Cos x) - ' . b )c c

7. De mo s t r a r q ue s i h e s c o n t i n ua , f y g s o n d e r i v a bl e s .g ( x )

Y F ( x ) h ( t ) d t , e n t o nc es F ' ( x ) = h( g ( x ) ) #g' ( x ) - h( f ( x ) ) #f ' ( x )f ( x )

8. Ha l l a r A ' ( x ) s i A( x ) Ll d u.1+u*.'3

1 . 1 7 P R I MI T I V A DE UNA F U NC I ON ( I NT E GR AL I NDE F I N I DA ) .De f i n i c i n . Una f unc i n F s e d en omi n a una pr i mi t i v a de f en[ a, b3 s i y s o l o s ia) F e s c o n t i n ua en [ a , b ]b) F ' ( x ) =f ( x ) p ar a t odo x en ( a , b) .E j e mp l o 1. F ( x ) =x E +l e s una p r i mi t i v a de f ( x ) =2x en [ - 5 . 5 ] ;t ambi n l o e s en ( , , +co) ; p ue s F ' ( x ) =2 x y F~ e s c o n t i n ua . P e r oF ( x ) =x a +2 e s t a mb i n u na p r i mi t i v a d e f ( x ) =2x en [ - 5 , 5 ] . Enge ne r a l F ( x ) =x +c e s u na p r i mi t i v a de f ( x ) =2x en [ - 5 , 5 ] y a s i s epue de a f i r ma r q ue s i F ( x ) e s una pr i mi t i v a de f ( x ) , e nt o nc e s p ar ac ua l qu i e r v al o r de c , F( x ) +c e s una p r i mi t i v a de f ( x ) .E j e mp l o 2. F ( x ) S en x+c e s u na p r i mi t i v a d e f ( x ) = Co s x en( OD +


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El p r i me r t e or e ma f u nd ame nt a l del c l c u l o e s t a bl e c e q ue s i f e s

c ont i nua en [ a , b] ; s e v er i f i c a que F ( x ) f ( t ) d t es unaa

pr i m t i v a d e f en [ a , b] . Es t o n os da una r e c et a pa r a c o ns t r u i rpr i m t i v as y a de m s a F ( x ) s e l l a ma i n t e g r a l i n d e f i n i d a d ef .El s i g u i e n t e t e o r e ma mu e s t r a , c o mo s e p ue de us a r c u a l q u i e rpr i m t i va pa r a ha l l a r l a i nt egr a l i nde f i n i da de una f unc i n f yse c ono c e c omo el S e g u n d o t e o r e ma f u n d a me n t a l d e l c l c u l oy di c e a s i :Sea f c o n t i n ua en [ a , b ] . Si F e s u na p r i mi t i v a de f en [ a , b ]

b be nt o nc e s f ( x) d x F ' ( x ) dx = F ( b) --F ( a) = F ( x )b

aDe mo s t r a c i n .Sea B( x ) una p r i mi t i v a c u al q ui e r a de f en [ a , b] , e nt o nc e s B( x ) :

f ( t ) d t . Co mo F e s t a mb i n u na p r i mi t i v a d e f s e t i e n e qu e.

G( x ) - F ( x ) = c ( E j e r c i c i o ) ; y as i G( x ) =F ( x ) +c f ( t ) d t ;a

1 u e go G( a ) =F ( a ) +c- ! 0 y a s i c : - - F ( a ) y F ( x ) - F ( a ) : f ( t ) d t. dea

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bdonde s e c o n c 1 u y e q ue F ( b ) - F ( a ) ~ f ( t ) d t

La f r mu l a es v l i d a s i a >b ; p ue sF ( b ) - F ( a ) a

af = - [ F ( a ) ~F ( b ) ]

Ej empl o 1. Ca l c u l a r x d x

So l uc i n.La f u nc i n F ( x ) = ( x 2 / 2 ) +c e s u na p r i mi t i v a de f ( x) =x: e n t o n c e s


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Solucin.

F ( K ) = ( 2 ; 3 I / 3 ) - B X + C e s u na pr i mt i v a d e f ( X ) =2x a - 8, l ue go

( 2x=*~8) dx = F ( x )- 1

C 2 ( 2 7 / 3 ) - 2 4 +c 3 - [ ( - 2 / 3 ) +8+c 3 = - 40 / 3

Ej empl o 4 . Ca l c u l a r ( x - 1 ) ( x +2) dx .

So l uc i n,( x- 1) . ( x +2 ) = x s e+( x 2 ) = f ( x ) , l u eg o F ( x ) = ( x 3 / 3) + ( x=V2) ~2x+c es u

1pr i m t i va de f ( x ) y as i f ( x ) d x - F ( x ) - . 10/ 3,

Ej empl o 5. Ca l c u l a r ( x - 1 ) l / 2 d

So l uc i n.Una p r i mt i v a d e f ( x ) =< x- 1) * e s F ( x ) = ( 2 / 3 ) . ( x - 1 ) 3- ' +c , l ue go

f ( x ) d x = F ( x ) 2/ 3.

E j e mp l o 6 . ( 2 x ) e* " d x0

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Solucin.

F ( x ) =ew* +c es u na p r i mi t i v a de f ( x ) =2 xe " * , l ue go

f ( x ) d: F ( x) e + c = e - 10

1 . 18 E J E R C I C I OS .Cal c ul ar l as s i gu i e nt e s i n t e gr a l e s :

(> 'V3 / 2 - wl / 2 ) d X 0

0( ) d x .X + l

( x ( x=*- -1) ) d:0

4. ( - x * +4) d :

( 3x =- x +l ) d ; Senxd;I T / 6

x j d; 8, ~4 | dx

l i ed: 10. 3 | d ;

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11. ( e - " +( 1/ x ) ) d x .1

.1.0

1 3 , ( x - 5 ) SOdx

1 5 .

0.1

0d xl + >

1 2 ,

1 4 ,

16 .

4( e2KH- 3+2)d ; - !

0

( ( 1/ x ) + ( 2x+l )11

0

dx - dx .( 1 - x a )

Ob s e r v a c i o n e s . De b i do a l a c one x i n e nt r e p r i mi t i v as e. i nt e gr al es d e f i n i d a s q ue p r o po r c i o na n l o s t e o r e ma s f u nd ame nt a l e sdel c l c u l o ; s e ha h ec h o c o s t u mb r e u s a r s i g no d e i nt e gr a l pa r adenot ar p r i mi t i v as . P ar a po der d i s t i ngu i r e s t a s de l asi nt egr a l es d ef i n i da s , no s e es c r i be n l i m t es de i n t e gr a c i n .Si no s e t i ene n i ngn i n t e r s pa r t i c u l a r en el i n t e r v a l o [ a , b ] ,s i no q ue s e d es e a r e s a l t a r q ue F e s u na p r i mi t i v a d e f s ees c r i be s i mp l e me nt e .

f ( x) dx = F ( x ) +c =G( x ) s i y s o l o s i G' ( x ) =f ( x ) y a s t a spr i m t i v as s e l l amar n i n t e gr a l e s i nde f i n i da s .En g ene r a l no s e da e x pl c i t a me nt e el d omi n i o de f , S i e mp r es e s u p o n e q u e s e h a e l e g i d o u n i n t e r v a l o a d e c u a d o e nel c u a l f e s i n t e g r a b l e .Al i gual q ue en l a i n t e gr a l e s d ef i n i d as , no i mpo r t a c ua l e s el

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s mbol o e mp l e a do pa r a l a v a r i a b l e d e i n t e gr a c i n

E j e mp l oa)

b)

c )

( x * ) d x = ( x * / 4 ) +c = G ( x) ; pues G' ( x) =

S e nt d t = - Co s t +c

e" d = e" +c .

1 . 19 A L GU NA S P R OP I E D A D E S DE L A I NT E G R AL I N DE F I N I DA .

1. 19. 1

1. 19. 2

ddi f ( x ) dx = f ( x ) ( t e or e ma f u nd ame nt a l ) .

ddx ( f ( x ) dx ) = f ( x ) +c , p ue s f ( x ) +c e s u na p r i mi t i v a

f ' ( x ) .1 . 1 9. 3 S i c e s un n me r o r e al e n t o n c e ;De mo s t r a c i n .

c f ( x ) d x = c f ( x ) dx .

ddx ( c f ( x ) d x ) = c (d x f ( x ) d x ) = c ( f ( x ) ) ; 1uego

c f ( x ) d x = c f ( x ) d x .1 . 19 . 4 Cf ( x ) +g ( x ) ] d ; f ( x ) d x + g ( x ) d:De mo s t r a c i n .

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f ( x) d x + dg ( x) d x ) f ( x ) d x +f ( x ) +g( x ) , l uego

dxCf ( x ) +g( x ) ] d

dxf ( x ) d x +

g ( x ) d xg ( x ) d x .

Tambi n s e p u ed e ob s e r v a r u na r e l a c i n e n t r e l as i n t e gr a l e si nde f i ni das y de f i n i da s as i : bF ( x ) +c ba ( F ( b ) +c ) - ( F ( a ) +c ) =F ( b ) - F ( a ) =F ( x )

ba f ( x ) d :a

f ( x ) d x a .Es t o mu es t r a q ue pa r a c a l c u l a r un a i nt e gr a l d ef i n i d a s e p ue dec al c ul ar l a i nde f i n i da y al f i nal c o l o c ar l e l os l i m t es .

1 r - 1E j empl o 1 . e ** d x = e " d x r e> + c00 \ J J

e - 1 .0Por l o a nt e r i o r n o s d ed i c a r e mo s m s a c a l c u l a r i n t e gr a l e si nde f i n i das , que de f i n i da s .E j e mpl o . 1 Ca l c u l a r ( 3x+5) di f ( x ) d x

Sol uc i n,

l a f ( x ) d x ( 3 x ) d x + 5d x x d x + 5 xdi

C( 3/ 2) x=+c] +[ ( 5x + b) 3 = - x=2+5x + k - F ( x ) , d o n de k =c +b .

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Ej empl o 2 . Ca l c u l a rSo l uc i n.

( S e c 2u+Co s u ) du f ( u ) d u .

Una p r i mi t i v a d e f ( u ) e s T an u+S en u+c = F ( u ) , p ue s F ' ( u ) =f ( u ) dFdu f ( u ) q ue en t r mi n os de d i f e r e nc i a l e s s e t i e ne qu e dF =f ( u ) d uy as i F ( u) = Tanu+Senu+c= ( S e c 2u+Co s u ) du S e c 2 udu + Cosudu-

Ej empl o 3 . Ca l c u l a rSol uc i n,

( + 3 ) 2 . x a d ;

( +3 ) 2 . x 2 d xD ( + 3x * + 9 x a ) d :4 dx + 3 x *dx +

9x=dx = + _ x B + 3 x 3 +c.28 5

1 . 20 E J E R C I C OS .J us t i f i c ar l as s i gu i e n t e s i gua l da de s

x n d x - + c , s i n - 1n + 1 I n I x I +c

a" du = +c; a>0, a l 4,l naSec udu = I n I Secu+Tanu I +c 6 .

T a nudu = L n | Se c u | +c

Cscudu = 1 n I Cs c u - Co t u I + c

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7.

9.

11,

Cs c ^t d t = - Co t t +c .

Cs c z C o t z d z = - Cs c z +c .dx Ar c T an ( ) +ca a^+x^

- Co s ax+c = - ' 4Cos2>; +c

8 ,

10.

12.

T a nt S e c t d t = S e c t +c .

Ar c Se n ( ) +ca2Senx Cosx dx = Se n 2 x +c

1 . 2 1 ME T OD OS DE I N T E G R A C I ON .1 . 2 1 . 1 S U S T I T U C I ON ( C A MB I O DE V A RI A B L E ) .Si g' e s c o nt i n ua en [ a , b ] y f e s c o nt i n ua y t i e ne una p r i mi t i v aen un i n t e r v a l o qu e c o n t e ng a l os v a l o r e s de g ( x ) p ar a x en [ a , b ] ,e nt o nc e sb g ( b)

f ( g ( x ) ) . g' ( x ) . dx =

exi s t anDe mo s t r a c i n .

f ( u) du , s i e mpr e que l as i n t e gr a l e sg ( a )

Sea u =g ( x ) y F u na p r i mi t v a d e f e n un i n t e r v a l o q ue c o n t e n g al os v a l o r e s d e g ( x ) pa r a x en [ a ^b ] e nt o nc e s po r l a r e gl a d e l ac a de na s e t i e n e

d dF du du( g ( >) ) = F ( u ) = = f ( u )dx dx du dx dx f ( g ( x ) ) . g' ( x) pa r a

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c ada x e n [ a , b ] . De e s t e mo d o , F ( q (;;)) e s u na p r i mi t i v a d ef ( g ( x ) ) . g' ( x ) en por l o t a nt ob g ( b)

f ( g ( x ) ) . g' ( x ) . dx - F ( g ( x ) ) = F ( g ( b ) ) - F ( g ( a ) ) : f ( u ) d u .g ( a )

De l o a n t e r i o r s e p ue de c o n c l u i r t a mb i n q uef ( g( x ) ) . g ' ( x ) . dx = F ( g ( x ) ) +c f ( u ) d u = F ( u ) +c s i u =g ( x ) q ue

es l a f r mul a del c a mb i o d e v ar i a bl e par a l as i n t e gr a l e si ndef i ni das5

Ej empl o 1. Ca l c u l a r t . ( t a - 4 ) . dt .So l uc i n.Sea u=t a - 4 = g ( t ) ; d e mo d o q u e d u=2 t . d t .Cuando t =2 , u- ~0 y cua ndo t . =5, u=21 , l uego po r e l t eo r ema ant e r i o rs e t i e ne q u e5 21

t . ( t a - 4 ) i a t . d t = "


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Solucin.

Sea u=x 3 +l ; d u =3 x 2 . d x y as i u =l c u an d o x=0 y u =9 c u an d o x=2 ,2 9

e nt o nc e s x 3 . ( x 3 +l ) 1 / 2 . d x U du = u"90( 27- 1) 9 y ( x

3 +i ) . d x , u~ - +c ( x 3 +lT I / 2

Ej e mpl o 3 . Ca l c u l a r Co s 2 x . S e nx . d x0

So l uc i n .

Co s 2x S e n x d x u3 - ( C o s x ) 3u 2 d u = - +c = +c ; s i e nd o

u =Cos x ; y d u =- S en x d xu / 2 - ( Co s x ) 3Co s 2 x . S e n x . d x = +c

0

T I / 2 = 1/ 3.0

6 >-E j e mp l o 4 . Ca l c u l a r d x .- x+1S o l u c i n.Se hace u=x 3 x+.L; c l u=( 3x= L ) . d x ; LI=1 c u a n d o x v a l e 1 y u=7 cuandox =2 , a s i q u e

7 5


24/60

6; d xA f\

% T V ..... 1A JL dx duu

2 1 n I u 21 n7,

Ej empl o 5. Ca l c u l a r 1 ( 1 n x ) d x .So l uc i n.

d xSea u =l n x; d u = y as i ( 1 nx ) d: u 1 / 2 d u u 3 / 2 +c

( I n ;Ej e mpl o 6 . E v a l u a r e- ~dx

So l uc i n,1 p 3 Sea u= ; du= . dx e nt o nc e s e u d u

-.1 - 1 . e , a+c = e 3 / " +c y a s i - dx e 3 X K+( ~

Ej emp1o 7 . Ca l eu1 a r e- 3 - dx .e 3 M + lSo l uc i n. duSea u=e 3>


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Ej empl o 8. Ca l c u l a r e " ( e" + l ) x y a . d>i .Sol uc i n.Sea u=e K +l ; d u =e w. d x y a s i e ( e + i ) i / s , d >: u 1 / B . d u5 u ' ^+c ( e K +l )

Ej e mpl o . 9 Ca l c u l a rSo l uc i n.

S e n3 x . d x .

duSea u=3x ; du=3 . dx , dx y as i Sen3xdxCosu+c , Cos3x+c .

Senudu =

E j emp1o 10 . Ca l cu1 a rSo l uc i n.

( x + 1 ) . d;

Sea u=x+l ; du=dx ; as i ( x +l ) 1 / a . d x ( u - 1 ) u 1 / 2 . d u

( u 3 / 2 - u 1 / a ) . d u u U 3 ' = + C = ~( x +l ) - ( x +1 ) 3

V E R I F I C A R L A S S I GU I E N T E S I GU A L DA DE S

1. 3x ( l - - 2x2) t / a . d x ( l - ~2x2 ) 3' ' 3t +c

) i / 2 . d x ( l - 2 x a ) 3 / 3 +c

7 7


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- dx = X I n ( x+1 ) +c: + l( e " C o s eM) dx = Sen e H+c

1+ = _ ( 1 + y * ) 3' ' : 2 + c .6

( X+ 1 ) dx = ( >; a+2>; - 4) 1/ 2+c .( x a +2 x - 4 )( 1 +x 4 ' ' a ) a dx = - ( l +x A ' =) 3 +c

e wx a e M dx = +c

e" r - " =" Sec=2xdx i Tn2x +ce- 1dx = Ar c T a n ( e 2 M) +c .

Ar e l a n;1 +K = dx = - ( A r c T a n X ) a +c .

( 4+3e M ) 1 / 2 e " . dx = _ ( 4+3e" ) 3 ' a +c

I n ( I n ( X ) )x l n ( x )

9( I n ( I n ( X ) ) a +c

C o s x e a - n Kd >: = e e - r >w+c .

7 8


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15.

16.

17.

18 .

19.

( Se nx +Co s x )( S en x- Co s x ) =

[ ( .l +x 2 +( l +x =) 3 / 2 ] -

( S en x - Co s x ) 2 / 3 +c

= 2 ( l +( l +x 2 ) 1 / 2 ) 1 / 2 +c

( 2 - l ) i / 3 d2 = - ( Z- l _ ( z - n ^ ^ + C7 4( l n ( x +( x 2 +l ) 1 / 2 ) =

(1+ y * f ( l n( x +( l + xa )

( 1 - x * ) dx -- - Ar c S e n ( X a ) +c

Cal c ul ar l as s i gu i e nt e s i nt egr a l es ,

20. Sen xCos- d x. 2 1. d x, 1+e"( 1+Senx ) dx . 24. ( 1 - e * ) A ' a . d

1+e"1

dx .

x ( x OB, + l )1 . 2 1 . 1 . 1 I NT E G RA L E S DE F U NC I ON E S T R I GON OME T R I C A S .

L a s . i nt egr al es d e l a f o r ma ( S en " x . Co s mx ) d x ; m y n N.Es t e t i po de i n t e gr a l e s s e r e s o l v er n de l a f o r ma s i gu i e nt e :a ) s i m=0 ; n=y3; ent onc ess i n e s pa r o i mp a r .1. S i n e s i mp ar e n t o n c e s

S en " x d x s e r e s o l v er c o n s i d er a nd o

Se n" x dx = ( S e n " - i x . Se nx ) d x

/ v


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[ ( Sen2 X ) > c e n ] d [ ( l - Co s 2 x ) > ' . Se nx l dx =( 1 - u 2 ) < >/ , = . du ; ( s i e nd o u=Co s x ) ; q ue e s f c i l de r e s ol v er .

Ej empl o 1. Ca l c u l a r Se n3; ; dx .

So l uc i n. Se n : sxd; ( Sen2> . Sen:- : ) dx ( l - Co s =x ) Senx d;u=Cosx

( l - u 2 ) d u = 1 1( u 2 - l ) du = u 3 - u+c = - Cos 3 x Co s x +c"T "TEn f or ma a n l o ga s e c a l c u l a l a i n t e g r a l de l a f o r ma

Co s mx dx ; c ua ndo m e s i mpa r y n=0 .2. S i n es pa r y m=0 ent onc es Sen" xd x ( S e n 3 x ) n / 2 d x

r ( l - Cos 2x dx q ue e s f ci l d e r e s o l v er .

Ej empl o 1. Ca l c u l a r S en * x d)So l uc i n.

Sen*xd: [ ( l - Cos 2x ) :( S e n 2 x ) 4 / 2 . d x = dx =

14 ( l - 2 Co s 2 x+Co s a ! 2x ) dx =

l +Co s 4 X( l ~2 Co s 2 x + ) d:

8 0


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1 1 .1 .1- x - S en 2x + . x + S en 4x +c4 4 8 32Co s mx d x Cu a nd o m e s pa r s e c a l c u l a en f o r maa . i nt egral i

anl oga.b ) m y n 4=0.1. Si m=n y m, n s on i mpar es ent onc es

Sen"x . Co s " x d; ( S en xCo s x ) " d ;

S e n x . Co s mx d x =( Sen2x

1 ( S e n u ) " 1On On+l S e n " u d u q u e y a s e s a b e r e s o l v e r .

La i nt egr al a nt e r i o r t a mb i n s e p ue de r e s o l v er a s i :Sen"x Co s mx d x = Se n" x C o s " x dx

i LCo s " x ( S en 2 K )


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i. 1 . 6

S e n 3 u d u ; q ue y a f ue r e s ue l t a . t a mbi n s e p ue de r e s o l v e r d e l a f o r ma s i g u i e n t e :

3 e n3 C o s 3 x d x Co s 3 x SenxS en xd x o s 3 x ( 1 Co s ) . S e n x d xl f COS; u 3 ( l - u a ) d u ( u - u 3 ) du u*4 +c61 1- Co s ^x - C o s 4 x + c .6 4c ) Si m- n y m y n pa r e s e n t o n c e s

S e n " x C o s mx dx Se n r , x . C o s " x d x ben^x ) d x ( S e n 2 x ) " d ;que y a f u e r e s u el t a ,. a i nt e gr a l

Se n n x C o s mx d ;

S e n x C o s x d x s e p ue de r e s o l v e r t . ambi . n a s :

Se i Vx ( Cosx ) n / 2 d x Sen" x ( l ~Senx ) n / 2d>;que y a s e t r a b a j .Ej e mpl o 1. Ca l c u l a rSo l uc i n.

Sen " x C o s x d x

Sen4>: Cos4; - ; d ;

Sen^udu

( S e n xCo s x ) 4d;- = Sen2x( ) 4 d ; 116 .. Sei T*2xdx =uSe n2u u Se n4u+ _ + +c )4 8 32

8 2


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.1 2x Sen4x 2x SenBx ( - + + +c ) .32 4 4 S 32d ) S i . m n .1. Si n e s i mp ar y m e s pa r e n t o n c e sS e n n x C o s mx d x S e n x S e n >~* x C o s m x d x =

Senx ( S en a x ) >/ 2 . Co s mKd >; = S e n x ( l ~ C o s = x ) . C o s m x d :

( L - u a ) < r , _ a - >. u du s i U - C O S A . E n f o r ma a n l o g a , si . n e s p ary m e s i mp are) Si m y n s on i mp a r e s d i s t i n t o s e nt o nc e s

Sen r >xCos mx d ; S e n " " 1 x C o s m xSenxd :

(Sen2>; ) ~1 1 / 2Co 5mx Se n; ; d ; ( l - Co s 2 x )


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f ] Si m y n s on e n t e r o s pa r es po s i t i v o s d i s t i n t o s e n t o nc e

Se nxCo smx dx ( S e n

2x )

n / 2. CD5

m; ; d ; ( l - Cos2; - : ) n / 2 . Co s m>! d;

I n t e gr a l e s de? l a f o r maEj empl o 1. Ha l l a r

T a n n x dx . nN,(T a n " x d x .

So l uc i n.

Tan" xdx = T a n n ~a x T a n a x d x = T a n n ~2 x ( S e c 2 x - 1 ) d;T a n n - 2 x S e c 2 ; T a n n ~2 x d x u n ~2 d u Tan"- =>; dx

n- 1 T a nn - 2 x d x - T a n " " 1 n- 1 T a n

n - a x d x s i n l ; e s t a s ec o no c e c o mo una f r mu l a de r e duc c i nSi n =l e n t o n c e s " a nx dx = n I Co s x I +cEj empl o 2. Ca l c u l a r T a n3 x d ;Sol uc i n,

T a n 3x dx T a n 2 x T a n x d x ( S e c a x - 1 ) T a n x d x = T a n x S e c 2 x dx

Tanxd: udu u-Tanxdx = T a na xT a nx dx = +l n | Co s x | +c

L as i n t e gr a l e s d e l a f o r ma Co t " x dx n eN s e c a l c u l a n en

8 4


33/60

f or ma anl oga.I nt egr al es d e l a f o r ma Se CxdK; nel M

a) Si n es par ent onces 5 e c x d ; S e c " ~ax . Se c axdx

[Sec ax)


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b) S. i n e s par e n t o nc e s T a n mx S e c " x d x

T a n x S e c n - Z x S e c a x d i T a n mx ( 1 +T a n a x ) / 2 S e c 2 x d :

u m( l +u a ) t n - 2 ) / 2 d u s i u =T an x; y e s t a i nt e gr a l e s f c i l d er es ol ver .c ) Si m e s i mpa r e nt o nc e s Ta nmxSec r >x dx T a nm ~x x T a n x S e c " x d :

Tan- x x S e c ~ x T a n x S e c x d x (Tan2; - ; ) ' a Se c r , - : L x T a n x S e c x d x

( S e c ax - 1 ) 5 2 B e c n - 1 x ( Se c x T a nx ) dx ( u a - l ) < m " 1 > / 2 u " " 1 d u s i .u - Sec xE j emp1o 1 . Ca l eu1 a r T a n 3 x S e c 7 x d So l uc i n,

T a n 3 x S e c y x d x " an2xS e c x S e c x T an x d x

( Sec 2x- ~1 ) S e C- x S ec x Tan xd ; ( u' - 2- ! ) u - 'd i ( u e - u 4 ) d u

u7" +c: Sec ^x Se c y x +c

d) Si . m e s p ar y n e s i mp ar T a n mx S e c " x d x s e r e s o l v e r por el

8 6


35/60

mt odo d e i n t e gr a c i n por p ar t e s .En f o r ma a n l o g a s e c a l c u l a l a i n t e g r a l d e l a f o r ma

IC o t mx S e c n x d x ; m y n e N.I nt e gr a l e s d e l a f o r maSol uc i n.

Sen mxCos nxd x .

Se s abe que Se nmxCos nx = [ Sen ( m+n ) >: +Sen ( m~n ) x ] y as iOr iSenmxCos nxd; [ Sen ( m+n ) x+Sen ( m- n ) x ]dx =1 Cos ( m+n) x 1 Cos ( m- n) x +c s i m( m+n) 2 ( m- n)

Si m=n se t i ene que SenmxCos nxd ; Sen n x Cosnxd:

Sen=nxz n +c , s i n==0 y s i n=0 ent onc es SenmxCos nxd x =

Cos mxSe nmxdx = - +cmEn f o r ma a n l o g a s e c a l c u l a n l a s i n t e g r a l e s d e l a f o r ma

S e n mx S e n n x d x y C o s mx C o s n x d x y a q u e

S e n mx S e n n x = [ C o s ( m- n ) x - C o s ( m+n ) x ] y2

Co s mx Co s n x = [ C o s ( mn ) x +C o s ( m+n ) x 3 .2

8 7


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E j emplo 1 . Cal c ul ar Sen2xCos3xdx.

S o l u c i n .Sen2xCos3xdx 1 1[ Sen5x+Sen( - x ) ] dx = - CosSx + Cosx + c

10 2

E J E R C I C I OS

Ver i f i car l as s i gui e nt e s i gual dades .T I / 2

I . Cost dt 815 Cot xCsc 3xdx Cs c 3x +ci0

1 1Sec * 7xdx = T a n37x+ _ Tan7x+c21 74. Csc xdx = - Cot x Co t 3x - _ Co t s x +c.55. Tan33xSec*3xd: 1 1 Tan"*3x+ Tan Sx+c

12 18

6. Co t 3 Csc x d x 1 1- C s c 7x + - Csc x+c .7 5Cos4xCos2xdx 1 .1 Sen2x + Senx +c4 12Ta n 3 / 2KSec 4Kd: - T a n / 2x + - Tan' / 2x +c5 9

9. Cal c ul ar l as s i gui ent es i nt egr al es

8 8


37/60

Co s 3 x d x b) Co s a xd : O Sen x d x d ) S e n x C o s s x d :

1 . 21 . 1 . 2 S U S T I T U C I ON E S T R I GON OME T R I C A S .Cuando un i n t e gr a nd o c o n t i e ne p ot e nc i a s e nt e r a s d e x y p o t e nc i a se n t e r a s de a l g un a de l as e x pr e s i o ne s ( a 2 - x z ) 1 / 2 , ( a 2 +x 2 ) 1 / 2 ,( x - a2) 1/ ' 2 a >0 , a me nu do e s p os i b l e el v a l u ar l a i n t e gr a l p o rHed i de u na s u s t i t u c i n t r i g omo me t r i c a , l a c u al n os da u nai n t e g r a l q ue c o n t i e n e f u nc i o n es t r i g o no m t r i c a s q ue a me n ud o s onf ci l es de r e s o l v e r . Ob s e r v e mo s l os s i g ui e nt e s c a s o s :Cas o 1. Cu a l q ui e r i nt e gr a l de 1 a f o r ma R ( x , ( a 2 - x 2 ) 1 / z ) d xdonde R e s u na f u nc i n r a c i o na l en l a s d os v a r i a b l e s x , ( a a - x 2 ) l / 2

se puede t r a ms f o r ma r me di a nt e l a s u s t i t u c i nx = a S e n t , - T I / 2 1 t


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a Js us t i t uc i n

Ej empl o 1. Ca l c u l a rSol uc i n.

S e n t ; =d x

( a) *( b) Cos t ; a>0 , b>0 .

Sea :


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( 4 - x 2 )que Cost -

Si s e qu i e r e c a l c u l a r l a i nt egr a l de f i n i da ( 4 - xa) 1/ at : i x , s ecal cul a l a i nt egr a l i nde f i n i da y al f i nal s e l e c o l o c an l osl m t es a s :

x ( 4 - x 3 ) =(4 -xa ) 1 / a d x = 2Ar cSe n ( x / 2 ) + +c ; a s i q ueIr 2x ( 4 - x 2 )( 4 - ; ; 2) 1 / a dx = 2 A r cSe n( x / 2 ) + +c 4 Ar c S e n 1

2TC ; t a mb i n s e p ue de c a l c u l a r a s :2 * ti/2 TI/2

( 4 - x a ) 1 / 2 dx = ( 4Cos 2 t ) d t = 2 ( l +Co s 2 t ) d t = 2rc =- u / 2 - t i / 2

T I / 2

r e/ 2( # Co mo x =2 Se n t ; c ua ndo x =2 , Se n t =l y a s i t =i t / 2; y c ua ndo x =Sent =- 1 y a s i t =- n / 2 ) .

x d xEj empl o 3. Ca l c u l a r

Sen2t .!(t + )

4- x 5B+ ( 4 - X a )So l uc i n.Sea x =2 Se nt , d x =2 Co s t d t , ( 4 - x 2 ) 1 / 2 = 2Cos t y as

4 - x 3 +( 4 - x 2 ) l / 22Sen t . 2Cos t d t4Cos a t +2 Co s t

' Se n t d tCo s t+-

9 1


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- l n | ' +Cost. | +c = - l n 1 ( 4 - X5* ) t y s t +c

R ( x , ( a a - ( c x +d ) 2 ) ^3 ) d x ,s a n , s i l a i nt e gr a l e s d e l a f o r mase p ue de r e s o l v e r en f o r ma muy s i mi l a r s i s e h ac e l a s u s t i t u c i no; +d=aSenu c x+d=aCos u; a>0.

dxEj empl o 1. Ca l c u l a r ( 8 +2 x - x a )Sol uc i n.Par a e s t e t i po de e j e r c i c i o p r i me r a s e c o mp l e t a el c u a dr a doper f ec t o a s i ; 8 +2 x - x a =8- Cx a - 2 x ) =8~ ( x a - 2 x +l - 1 ) =

dx8 - ( x a - 2 x +l ) +l = 9 - ( x ~l ) 3 y as i

dx( 8 +2 x - x a ) 1 / 3

; s e hac e ( x- l ) =3Senu, y de aqui s e t i e ne( 9 - ( x - l ) a ) A ' ax =3 Se nu+l , dx =3 Co s udu ; ( 9- ( x - - l ) a) * ' a = 3Cosu,l uego,

Ar cSen(

dx( 9 - ( x - 1 ) a )

3 C o s udu3Cosu du = u+c

x - 1


) +c y a que x - l =3 Se nu .2xdx

( 8 +2 x a - x * ) 1 ' a

Sx U A

( 8 +2 x a - x * ) ' ( 9 - ( X a - 1 ) = = ; s i s e ha c e u=xa- - l , e nt o nc es

9 2


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=f xdx y a s i 2 x d ;( 9 - ( x a - l ) 2 ) 1 / adu

( 9 - u =) A' = Ar c S en ( u / 3 ) +cx a - lAr cSen( ) +c*

Ej empl o 3 . Ca l c u l a r ( 5 - x=+2x) :So l uc i n.

5 - x 3 +2x = 6 - ( x - i ) 2 l ue godx

( 5 - x S! +2x ) 3' ' a!dx

; Y e n

l ugar de ha ce r l a s u s t i t u c i n x - l = ( 6) x / 2 S e n t s e p ue de h ac e r l a

s us t i t uc i n u =( x - l ) y a s i du=dx y ( 6 ( x 1 ) 35du

( - u 2 ) 3 ' 1 ( Cu a n do x =2 ; u =l , y c u a n d o x =3 ; u =2 ) . Y p a r a

c al c ul a r e s t a i n t e gr a l s e ha c e l a s u s t i t u c i n u =( 6 ) t / 2 S e n e y d eaqu i du = ( ) x ' ' Cos ed y s e p ue de r e s t r i n g i r 6 al . i nt e r v al o0


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de du ci r de l a f i gur a

Por l o t a nt o2 du

du 1 1 u= _ T a nO+c = +c y as i( 6 - u 3 ) 3 ' * 6 6 ( 6 - u 5 * ) * ' 2u +c( 6 - u 2 ) 3 / 2 6 ( 6 - u a 3 ( 2 ) 6 ( 5 )

Cas o 2. Cu al q ui e r i n t e g r a l d e l a f o r ma R ( x, ( a a +x 2 ) 1 ) dx, a>0:R f unc i n r a c i o na l en l as v ar i a bl e s x , ( a 2 +x 2 ) 1 / 2 ; s e p ue d et r a ms f o r ma r me d i a n t e l a s u s t i t u c i n > = a l a n u ( x = a Co t u )- n/ 2


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Solucin.

Sea x - 2 T a nu , e n t o n c e s d x =2 S e c a ud u y 4+x==4( 1+Ta n a u ) =4 S e c 3 u yasi d 2 S e c 2 u d u( 4Sec' - 2 )4+ xa)3ysa1 1 X- Sen u+c = 4 4 ( 4 + x = 2 )

' 2S ec 2udu8 S e c 3 u

.14 Cosudu =

+c , y a q ue T an u = y as ioSenu c o mo s e p ue de d ed uc i r de l a f i g ur a s i g ui e nt e


dx( aa+x=*)

x=aTanu; dx =aS ec 2 u d u ; a 2 +x 2 =a 2 S e c 2 u , as i qu e

( a 2+x=)a S e c 2 u d u

a Se c uS e c u ( S e c u +T a n u ) d u *

( Se c u+T a nu )( a 2 +x s ) 1 / 2 x

dz

I n

Secudu =

l n | z j +c = l n | S e c u+T a nu | +c =x ( a 2 +x 2 ) l / 2+c ; y a que? T a nu = y Se c ua

Co mo s e pu ed e d e du c i r d e l a f i g ur a s i g ui e n t e . ( * z =S e c u +T a n u)9 5


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CL

Caso 3. Cu al q ui e r i nt egr al de l a f or ma R( x , ( x 2 ~a 2 ) 1 / 2 ) d x , a >0 ;F< f unc i n r a c i o na l en l as v a r i b l e s x , ( x - a5*) i / ' SB, se puede

t r a ms f o r ma r med i a n t e s us t i t uc i n x = a Sec u 0ua yu ! u


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que a p a r e c e c u a n d o s e

s i mpl i f i ca ( i2

a2

) e s t o e s ( xa- a

a)

i / a= ( 4 Sec

au - 4 )

1Tt / 3

! | Tanu| = 2Tanu, l uego r ( x a - 4) i / ad>i # "2Tanu2SecuTanudu Sec u0T I / 30 =2( 3 ) i / a - 2n / 3 ,

TT / 3 T I / 3T a n a ud u = 2 ( Sec a u - l ) d u = 2 ( T an u- u )

0 0(# Como x=2Sec u; c uand o x=2, Sec u=l y as i u=0 y c uand o ;-== 4 ,Secu=2 y asi U = = T I / 3 ) .

dxEj empl o 2 . Ca l c u l a r x 3 ( x 2 9 )Sol uc i n.sea x=3Secu, donde 0


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( x a - 9) i ' ' a 3En c u a l q ui e r c a s o , S e nu = y Co s u = Par a e x pr e s a r u en t r mi n os d e x s e d eb en d i s t i n g ui r d o s c a s o s ;Cuando x>3 y c ua nd o x3, u=Ar c Sec ( x / 3 ) . Y c uand o x


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5.( 2x - x a ) * ' a

dx- A r c S e n ( x - l ) -- _ ( x + 3 ) ( 2 x - x a ) + c

( X3 ) +c( 4 x a - 2 4 x +2 7 ) 3 / / a 9 ( 4x a - 2 4 x +2 7 ) x ' a

6 . ( x 2 - 4 x +1 3 ) 1 / = I n J X- 2 +( x a - 4 x +1 3 ) A ' a | +c ,

7. ( x a - l ) 1 / 2 d x = ( 3) * ' =* - _ l n ( 2 +( 3 )

8 .

10.

r e - d x( 9_ e=i =

x a dx( Xa- - 4 ) =

dxx ( 4 x a +9)

Ar c Sen( ) +c,

( x 2 - 4 ) 1 / a + 2 1 n | x +( x a - 4 ) * ' a | +c

I n( 4 x a +9 ) x y a t ~3

+c

11 ' dx 128( 1 6 - x 2 ) 1 / 2 2 4 ( 3 ) A ' a .

( 27) J./-212 . X a ( x a + 9 ) 1 2 7

3 ) 4L 2( 6 - ( 12 ) ' a ) .

13. dx( Xa 4) *1 1 it Ar c Cos ( - ) - 2 3 6

9 9


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14.

15.

16.

1 7 .

18.

1 9 .

20.

i l .

1 S e c a x dx( 4Tan3) - ; J 3 "

e t d t( es t +f j gt + y p / a

d x

"fan;-;4 ( 4 - T an 2 x ) x

e t +4+c .

9 ( e 2 t +8 e t +7 ) 1 / : +c .;+6= Ar cSen( ) +c .( 28 12 x x a ) i 8

1 ( 2x+3) dx 1 13= ._ I n I 9 x a - 1 2 x +8 I + Ar c Tan (9 x a - 12x +8 9 18 >+c

( x+2 ) d:< ( 4 x - x s ) 1 / 2 + 4Ar cS en(

( l - x ) i / 2+3Ar c : Senx+c ,

) +c

','Qbec xA r c T a n ( ) +c .


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

Como en e l c l cul o de

f ' ( X ) g ( X ) dX + c ,

f ' ( x ) g( x) dx or i gi na s u pr opi a c ons t ant e,no hay r a z n pa r a c o ns e r v ar l a c o ns t a nt e ; e s d ec i r ;f ( X) g ' ( X) d X = f ( x ) g ( x ) f ' ( X ) g ( x ) d X

Es t a f r mu l a s e l l a ma r , f r mu l a de i n t e gr a c i n po r p a r t e s y n osper m t e ha l l a r l av i nt egr a l f ( x ) g' ( x ) dx ; c uando s e puede c al c ul ar

l a i nt egr al f ' ( x ) g ( x ) d ;En l a p r c t i c a n or ma l me nt e s e u s a u =f ( x ) ; d v =g ' ( x ) d x d u=f ' ( x ) d x ;v=g( x) y l a f r mul a de i nt e gr a c i n por p ar t e s s e e s c r i b e a s i :

udv = uv v du . L a f r mu l a d e i n t e gr a c i n po r p ar t e s , p ar ac al c ul ar i nt e gr a 1es de f i ni da s e s

udv = uv v d u y e l x i t o d e e s t a f r mu l a d e p e n d e d ea al a e l e c i n d e u y d v .Ej empl o 1. Ca l c u l a r x d x ( x + l ) " 1 / a d x .Sol uc i n.Hay v a r i a s f o r ma s pa r a l a e l e c c i n d e u y d v ; s e p od r t e ne r

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dv=( x+l ) ~x / , s t dx ; d v =x d x d v =d x . Co mo un a g u i a p r c t i c a , l ae l e c c i n d e d v e s d e t e r mi n a d a p or l o q u e l e s u c e d a a l a s e g u n da. i nt egr al d e * . S i s e - e l i g e d v =( x + 1 ) ' - x ' adx j u =x s e t i e n e qu ebu=dx : v=2( x + 1 ) A/ ' a ( x+1 ) ~A/ ' adx , l ue go x ( x+i ) - * ' =2dx

2x ( x+1 ) ( x +1 ) 1 / a d x = 2 x ( x + 1 - - (Obs r v es e q ue n o e s n e c e s a r i o u na c o n s t a n t e e n l a i n t e g r a c i n dedv. L a c o n s t a n t e a gr u pa da al f i na l del p r o bl e ma e s u na c o ns t a n t ec ol ec t i v a.Ob s e r v a c i n . Si s e s e l e c c i o na r dv - x dx ; u = ( x + 1) , v=x a t / 2;du- ( x +l ) - 3 ' = y X

a 1x ( x +1 ) ~ 1 / a d x s ( x+l ) - A/ ' a+ -2 4Y el p r o b l e ma r e s u l t a e v i d en t e ; pue s l a i n t e gr a l

4 .

( x + 1 ) ~3 / a d x .

: 2 ( x +l ) - 3 / 2 d xes m s c o mp l i c a d a q ue l a i n t e gr a l o r i g i n al .Si s e e s c o ge d v=d x , u =x ( x +1 ) ~ x , t a mbi n c o n d uc e a un c a l l e j ns i n s a l i da .Ej e mpl o 2 . Ca l c u l a rSo l uc i n.

x l n xd x ; n N.

1 , r\ .1.Sea u =l n x ; d v =x " d x ; du = dx y v = ; l u e gox n+1 x l nx dx

l n; 1 d x = 1 n : / r> i- X .n + 1/ r JL 1 1X A JL. dx = l nx - ( n + 1 ) n + 1 ( n + 1 ) ' dx

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]. n x ~+c .n+1 ( n +1

Ej empl o 3 . Ca l c u l a r x 3 1 n x d;Sol uc i n.Se a pl i c a r l a f r mu l a a nt e r i o r c u a nd o n =3 y as i

x* 1 Tx 31 nxdx = l nx- _ x 3 dx = I n x - +c .4 4 J 4 16

Ej empl o 4. Ca l c u l a r l nr ' xdx; n N,Sol uc i n.

1 n n _ 1 x d xSea u- =l nnx , du= n ; dv=dx, v=x -y as i l n" x d:xl nnx - n l n " ~1xdx = x l n" x - n l n n ~1 x d xLa a nt e r i o r f r mu l a s e c o n o c e c o mo u na f r mu l a d e r e d uc c i n p a r ac al c ul ar 1 a 1 n x d xE j empl o 5 . Ca l eu1 ar l nx d;

So l uc i n,Se a pl i c a r el e j e mp l o an t e r i o r c u a nd o n =l , as i que:

1 n x d x = x 1 n x x . x = x 1 n x - x +c .

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Ej empl o 6. Ca l c u l a rSo l uc i n.

x " e x d;

Sea u=xr , | 1 du= nx n - 1 d x ; d v = e K d x , v =e w y x n e Hd x =

x ne M - n L e Kd x ( f r mu l a d e r e du c c i n)E j e mp l o 7 . Ca l c u l a r - e' - ' dxSol uc i n.Se apl i c ar el e j e mp l o a nt e r i o r c u a nd o n =2 , a s i q ue

e" dx = x 2e >l - 2 x 2 - 1 e " d x . Aho r a s e c a l c ul a

x e" d x, e n f o r ma ant ol oga c u a n d o n =l ; e s d e c i r ;

x e Kdx = xe' e*' dx = xe^- e^+c , l uego x a e Md x; ae M - 2 [ x e" - e * +c ]Ej empl o 8. Ca l c u l a rSol uc i n,

Se n n xdx neN

i e r i nxd; S e n " " 1 x S e n x d x ; s e a u = S e n n ~1 x ; d u=( n - 1 ) S en n " 2 x C s x d x ;dv=Senxdx , v=- Cos x y Sen" xdx =- Co s x Se n - i x + ( n - 1 ) S e n " x Co s x Co s x d x

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- Cos x Sen " - i ; - ; + ( n - 1 ) Senr*~a - ; Cosaexd;

- CosxSen"- 1; - : + (n- . l ) Sen"- ax( l - Senax ) dx

- CosxSen" - 1; - ; + ( n - 1 ) Sen"- 2xdx - ( n- 1) Senn~axSenaxd;

- CosxSen" - * x + ( n - 1 ) Senn - axdx ~ ( n - 1 ) Sennxdx ; y a s i

Sen"xdx + ( n - 1 ) Senndx = - CosxSen" " 1x+( n- 1) Senn~axdx , per o,Sennxdx + ( n - 1 )r Sennxdx = n Sen"xd;- C os x S e n x + ( n - .1 ) Senn - a xdx; y as.i Sennxd;

CosxSen" - Ax+( n- 1) Sen" " 3( f r mu l a de r educc i n)

Ej empl o 9 .Como una apl i c ac i n al ej empl o ant er i or ca l cul ar qen2xdxSol uci n.

n=2; CosxSenx+( 2- 1)Sennxdx - CosxSenx+xdx = +cSen2x +c .

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Ejemplo 10. Ca l c u l a r x"Sen; ; dx nN.

S o l u c i n .

Sea u=x" , d u=nx n ~1 d x ; d v =S e n x d x ; v =- Co s x ; y x "Senxd;' Cosx + n x" ACo s x d x ( f r mu l a d e r e d uc c i n )

Ejemplo 11. Ca l c ul a r xSenxd:S o l u c i n . .

Por u na a pl i c a c i n de l a f r mu l a a nt e r i o r ( n =l ) s e t i e ne que:xSenxdx = - xCos x+1U x 1 ~1 Co s x dx = xCos x+Sen x+cEjemplo 12. Ca l c u l a r e K C o s x d x .S o l u c i n .

Sea u=e K , du=e" dx dv=Cosx dx ; v=Senx y e - Co s x d ;e" Senx- S e nx e Hd x ; a h o r a e MS e n x d x s e c a l c u l a por p ar t e snue v a me nt e , l ue go s e a u : =eM, d u =e K dx ; dv =Se nx dx ; v = - Co s x y

e NCos xd x = e K Se nx e HSe nx dx = e HS e nx - [ - e K C o s x + e wC o s x d x ]e HSenx+ewC o s x - e T o s x d x y as i e HC o s x d x + e HC o s x d x =

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e" Cosx dx = e MSe nx + e HCo s x +c ; l ue go e KC o s x d ;e*Senx+e MCo5x + b,Ejemplo 13. Ca l c u l a r Se c 3x dSol uc i n.Sec 3xdxr

S e c x S e c 3 x dx ; dv =Se c s exdx ; v=Tanx ; u=Sec ;

du- SecxTanxdx y S e c 3 xd x = Sec xTa nx T a n 2 x S e c x d x -

SecxTanx ( Sec 2 x -- 1) Sec x d x=Sec x Tan x Se c 3 xd x + Sec xdx =

SecxTanx S ec ' x dx + l n I S e c x +T a nx I , l ue go 2 Sec 3; ; d x =Sec xTan x +1 n I Se c x +T a nx | +c ; e n t o ne e s S ec ' x dx =SecxTanx+l n I Secx+Tanx +b.

Ejemplo 14. Ca l c u l a r Ar c T a n x d xSol uc i n.Sea u=Ar c Ta nx, du= dx y dv =dx ; v ~x y as i1 + x 2 I' xdx , 1xAr c Tanx - = x Ar c T a n x- l nl l +x l +c .1 +x 2 2

Ar e T an

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Mediante in te gr ac in p o r partus, demostrar

Cos nxdx Cos n - 1x Cosxdx = Co s n_ 1xSenxnn - 1

n Co sn - a xdx n^N

Sen" xCos mxdx Senn" 1 xCo5n _ 1x SenxCosxd:Sen"* 1xCos" ' - i m- 1+

m+nSen r , xCosmxd:

n - m; n , m= N .

m+n . SennxCo5m- 2xdx.- Senn~1xCo5m+i x n- 1+

n+m n+m CosmxSenr >~axdx ;

4 . Secxd;

Csc"xdx

TanxSec n _ 2x n-Sec n" ax Sec 2xdx = + - Co t xCsc n_ 2x n - 2+

n- 1 n - 1c - ec n_ 2xd x ; n==1

n- 1 n - 1 Cs cn~2xdx ; n l , n N.

Senn+Xxd>: :L Sen" x n

8 .

9.

Co S M+ L X' C5m+1xdx

Sen"" 1xCos mxdx

m Cos mx m ,1 Cos mx mn Sen"x n

Senn _ 1xdxCos"

Sen"" 1:I

( dv=Senxdx)

jsa m- 1ben' ( m- n) Senn~1x m- n ,

Cos m~axdiben'

; m di .

x"Cosxdx = x" Senx- n ! n" " 1Senxd!

10, Sen( l nx) dx = - - [ Sen ( l nx ) - Co s ( l nx ) J +c

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e WC o s b x d x

e* " wSenbxdx

( a 2 - x 2 ) " d x

bSenbx+aCosbxe"*


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S en 3xe Hd; = e M 1 0Se n2x +5Co s 2x +l ) +cSen; - ;( l - x a ) * ' a d x = x ( l - x 2 ) x 2d :( l - x a )

P5. ( x 2 +l ) ( Ar c T an x ) 2x (AreTan; - ; ) 2 dx = xAr cT an: + - l n ( l +x 2 ) +c

26. xe r c T a nM ( x 1 ) e r- c 'T n x +c( l +x a ) 3 / a 2 ( i + x=a)

27. I n ( x+ ( 1 + x58 ) * ) dx = x 1 n ( x+ ( l +x 2 ) l ) - ( 1 + x 2 ) * ' a +c .

28. S en ( x 1 / 2 ) d x = - 2 ( x ) 1 / 2 C o s ( x * ) +2 Se n ( x A ) +c .

29. e


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11 ana l a d es c o mp os i c i n de en f r a c c i o ne s p ar c i a l e s ,i '-*-!Es t a d e s c o mp o s i c i n s e e mp l e a p ar a h al l a r l a 2dx y par a e l l ose i n t e gr a . i n de pe ndi e nt e me nt e c a d a u na d e l o s f r a c c i o n e s q uef or man l a d e s c o mp o s i c i n , e s d ec i r

2d;- 1

' d:x+1 I n I x - 1| - l n I x +1l +c

Ter i c ament e, e s po s i b l e e s c r i b i r c ua l qu i e r e xpr e s i n r ec i ona lf ( x) ( Con f ( x ) y g ( x ) p o l i n o mi o s ) c o mo l a s u ma d e e x p r e s i o n e sg( x)r a ci ona l es c u y o s d e no mi n a do r e s s o n p o t e n c i a s d e p o l i n o mi o s d egr ado no may or que dos .Co nc r e t a me nt e , s i f ( x ) y g ( x ) s o n p ol i n omi o s y el g r a do de f ( x )es me no r que e l g r a do de g ( x ) e n t o nc e s s e s i gue de un t e o r e ma de

f ( x )l gebr a que = F a . +F =+. . . . +Fr , . ( 1) d o nd e c a d a F i t i e ne u nag ( x ) A Cx+Dde l as d os f o r ma s s i g ui e nt e s : s i e nd o( px+q ) m ( a x 2 +bx+c ) r>m n e n t e r o s p o s i t i v o s y a d em s a x ^+b x +c no e s f a c t o r i x a b l e en R,es d ec i r bS - 4ac


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f ( X )par c i al es d e u na e x pr e s i n r a c i o na l e s n e c e s a r i o q ue elg ( x )gr a do d e f ( x ) s e a me n or q u e el g r a d o d e g ( x ) y s i s t e n o e s e l

f ( x )c as o, s e pue de e x p r e s a r c o mo l a s uma de un po l i no mi o , m sg ( x )una f unc i n r a c i o na l , f o r ma que s e o bt i e ne e f e c t u an do l a di v i s i n

f ( x ) R ( x )es d ec i r : = q ( x ) + c o n g r a d o d e g ( x ) >g r a d o d e R ( x ) .g ( x ) g ( x )En e s t a s e c c i n s e e s t u di a r n e s t o s d os c a s o s .f ( x )11- y g r a d o d e g ( x ) > g r a d o d e f ( x ) .g ( x )

f ( x ) f ( x )a ) . Si d o nd e t o d os l o sg ( x ) ( a Ax +b A ) ( a 3 x +b a ) . . ( a x+br , )f a c t o r e s a x + b*. s on d i s t i n t o s pa r a i = l , 2. . . , n; s e pue de de mo s t r a r

que ex i s t e n c o ns t a nt e s r e al es n i c a s c i , c 2 , . . . , c n t al q uef ( x ) Ci c *g ( x) a x + bi a 2 x +b 2" f ( x ) rdx =g ( x )

, l uegoa n x+br ,

a i x + b j.C' a

a2>; + b :-d x +. a-, X+br ,- dx' 2x + lEjemplo 1. Ca l c u l a r x +2 x - 3-d x .

S o l u c i n .

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Calculo Integral 02 - Bernardo Acevedo Frias