Calculo Integral 03 - Bernardo Acevedo Frias

8/22/2019 Calculo Integral 03 - Bernardo Acevedo Frias

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( A+B) : : +( 3A- B) , l ue go A+B =2 y 3 A- B =1 , y a s i A =- y B =- , l ue go4 4

: + l dx =A 1 o-, A '

, 3/ 4d xx 1

5/ 4d x +3 l n x - 1 | + _ 1 n | x +3 | +c .

Ejemplo 2. Ha l l a r el v al o r d eSol uc i n.

5 x +3 dx

5 x+3 A B__ +

Cx3- 2x a- 3x x ( x - 3 ) (x +1) : + l y as i5x+3=A( x- 3) ( x+1) +Bx ( x - 3) +c x ( x+1) ; p ar a x =0; s e t i e ne3=A ( 0 - 3 ) ( 0 +1 ) +B0 ( 0 - 3 ) +C0 ( 0 +1 ) =- 3 A; a s i ~3 A=3 ; A=- l .Par a x =3 s e t i e ne 5 * 3 +3 =1 8 =A ( 3 - 0 ) ( 3 +1 ) +B ( 3 ) ( 3 - 0 ) +C 3( 3 +1 ) =1 2 C;

.18 3l uego 1 2C=1 8, a s i C= = P a r a x =- l ; s e t i ene12 2- 5+3=- 2=A ( - 1 + 1) ( 13) +B ( - 1) ( - 1- 3 ) +C ( - 1) ( - 1+1 ) =4 B, l u eg o4B=- 2; B= ' / , as i que 5 x +3 - dx 1 1- d x - - ' dxx + 1-l n | x | - l n | x+1 j + - l n x - 3 +cO O

Ejemplo 3 . Ca l c u l a rSo luciV .

dx

A B+ ; 1 =A( x +3 ) +B ( x ~3 ) , l u eg o s i x =3 ; 1 - 6 A; A l / 6 .

1 1 3


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si >; =- 3; 1 =- 6 B; B =- l / 6 , l u eg o d x 6 .d x i

3 6dx

1 1- l n | x- 3 | - n | x +3 j +c .6 6

_ f ( x ) f ( x )b). Si = ; g r a d o f ( x )


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4 A + 2 B + 0 7A+B+C=4 Y s .i se r es ue l v e se t i en e que A= l , B=0 C=3, l uego

3 1 3a +2x +4 1 0< +l ) 3 X + l ( X +l ) = ( X +1) ~ 1 (x +l ) : 5 y

' x a+2x+4 1(. + 1) 3E j empl o 2 . C a l c u l a rS o l u c i n . .

+ )dx = l n I x +l I - - ( x +l ) - =+c .+1 ( x + l ) 3 2

; dx( x3) :

A B(; -3) * ( x3)31 , l ueg o

A B ] ( x - 3 ) 3( x - 3 ) ( x 3) :A ( x - 3 ) +B ; l u e g o p a r a x =3 s e t i e n e q u e 3 =A ( 3 - 3 ) + B y a s i B = 3 ;par a h a l l a r A , s e l e d a c u a l q u i e r o t r o v a l o r a x ,na t ur a l ment e x ^3 , as i p o r e j emp l o x= 0 x = l ; v eamo s s i x = 0 ;0=A ( 0 - 3 ) +B=- 3A +3=0 y as i A = 1 . S i se t o mar a x= l se t i en e q ue1=A ( 1 - 3 ) +B=2A +3; - 2A=- 2 y as i A = l ; l ueg o

1 3( x- 3) =

xdx

A B+

( x - 3) 2

( x - 3 ) ( x - 3) (x - 3)1 3

y as i

) d x = l n I x - 3 I + +c( x- 3) ( x - 3) :8

E j e m p l o 3 . C a l c u l a r ( x - 3 ) *

1 1 5


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Solucin,

x dx 3. ( I ni x 3 I )( x - 3 ) a ( x + 3 )8 3 3

= ( l n 5 - _ ) - ( l n 2 - - ) .5 5 25

c ) . C o m b i n a c i n d e l o s 2 c a s o s a n t e r i o r e s .

E j em p l o 1 . C a l c u l a rf(>x-l) d

3 ( 2 - x l )

So l uc i n.6x -l A B C D= _ + + + y a s i[ ( 2 x - l ) , S2 w 3

C a s o b ) C a s o a )A B C D

6x-l = x 3 ( 2 x - 1 ) ( + + + ) , l u e g o 6 x - l =vs , , 3 r> w 1A A jl. A J.

A ( x = ) ( 2 x - l ) + B x ( 2 x - l ) + C ( 2 x - l ) + D x 3 y a s i p a r a x = 0 s e t i e n e1- 1 = C ( - 1 ) , c = l. P a r a x=' s e t i e n e q u e 3 - l = 2 = D . - , D = 1 6 .8

P a r a x =l s e t i e n e q u e 5 = A ( 2 - 1 ) + B ( 2 - 1 ) + C ( 2 - 1 ) + D = A + B + C + D .P a r a x =- l s e t i e n e q u e - 7 = A ( - 2 - 1 ) + B ( - 1 ) ( - 2 - 1 ) + C ( - 2 - 1 ) + D ( - 1 ) 3 =- 3 A + 3 B - 3 C - D ; l u e g o s e t i e n e e l s i s t e m a s i g u i e n t eA+ B + C + D = 5l 3 A + 3 B - 3 C - D = - 7 A + B + 1 + 1 6 = 5

y as i y de a q u i B=--4 y A = - 8D= 16 3 A + 3 B 3 1 6 = 7C=1

Ah o r a p o r e l o t r o m t o d o p a r a h a l l a r A , B , C , D ; s e t i e n e6x - l = A x a ( 2 x - 1 ) +B x ( 2 x - 1 ) + C ( 2 x ~ l ) + D x 3 =

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( 2A+D ) x 3 + ( - A+2B ) x 2 + ( - B+2C ) >-C y as i2A+D- 0- A+2B=0 y r es o l v i endo es t e s i s t ema se t i ene C=l , D==16, , B=- 4 y- B+2C=6 A=- 8; l uego- C=- lr ( 6x- i ) d x>; 3( 2x- l )

- 8 4 1( _ + 162x 1 )dx =4 1-81 n | x | + - + 81 n | 2x - l | +c =

81n +4; - x~a+c

d) . Cu ando el d en om n ad or t i e ne f a c t o r e s c u ad r t i c o s n of ac t or i z abl es , ( b3S - 4ac


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S o l u c i n .4x Ax +B Cx +D+ y de aq ui 4x =( x a+l ) ( x a +2x+3) x a +l x a +2x+3

( Ax +B) ( x a +2 x +3 ) +( C x +D) ( x a +l ) =( A+C) x 3 +( 2A+B+D) x a +( 3A+2B+C) x+( 3B+D) y as i

^A+C=02A+B+D=0 Re s o l v i e nd o e s t e s i s t e ma r e s u l t a qu e3A+2B+C=4 A =1, B=l , C=1, D= - 3 ; l ue gol v3E


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Ej empl o 1 . Ca l c u l a r ( x a +4) 2Sol uc i n.

y a Ax+B Cx+D+ , e s t o c on d u c e a( x2+4 ) 2 x a +4 ( x 2 +4 ) 2x 2 = ( A ( x ) +B) ( x2+4) +( Cx+D) =Ax 3 +Bx a + ( 4A+C) x+ ( 4B+D) y

A=0; B=l ; 0=4A+C; 0=4B+D; y de aqu i r es u l t a que A=0, B=1, C=0y D- - - 4. P o r c o ns i g ui e nt e .

.1 4dx( x 2 +4) 2x 2 d ;

( ) d x . A h o r a( x 2 +4 ) ( x a + 4 ) 2x=2Tan6

( ; ; 2+4 ) :2 S e c 2 e d 6

( 4 Ta n 2 e +4) 2 16 ,2 S e c a 6de 1

Se c ^ e 8 Co s2 e d e

1 1( l +Co s 2 6) d 6 = ( 6+' Sen2$) = ( e +S en OCo s ).16 1 6

1161 x 2 ( Ta n ~x ( x / 2) + .16 ( x a +4 ) 1 ' 2 ( x 2 +4 ) y ' a ) por l o t a nt o s e t i ene

que ( x 2 +4 ) 21 1 1 x- T a n - " 1 ( x/ 2) - 4[ T a n- * ( x / 2 ) +2 16 8 x a +4 ] +c

- T a n " 1 ( x / 2 ) -4 2 ( x a +4) +c

f ) . Co mb i n ac i n de t o do s l o s c a s o s a l g n os d e e l l o s ,x +3Ej empl o 1 . Ca l c u l a r d x.; 4 +9x 2

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S o l u c i n .

;+3 x + 3; +9xa x a ( x a +9) x x a x a +9A B Cx +D + + y as i x +3X X a . . r . nAx ( xa+9) +B( x a +9 ) + ( Cx+D ) X a y A+C=0; B+D=0; 1=9A; 3=9B, por l ot ant o A=l / 9 , B =l / 3 ; C=- l / 9 ; D=- l / 3 y

; + l dx =a ( x a +9 )1/ 9 1/ 3( +

1 1- l n| x | -9 3

1/ 9 1/ 3 ( x / 9 ) +( 1 / 3)( + ) d x =x x a x a +9) d;1 8 ( x a +9)

1( x a +9)

1 n I x a +9 I - - T a n " " 1 ( x / 3 ) +c18 91 x a 1 1 l n( ) - - x - * - _ T a n - 1 ( x / 3 ) +c ..18 X a +9 3 9

f ( x )2) . y g r a d o de g ( x ) < g r a do f ( x ) .g ( x )Ej e mpl o 1. Ca l c u l a rSo l uc i n.

- 2x1V -i.. T x./ + dx

Obs r v e s e que e l g r a do del nume r a do r e s ma y o r que el g r a do delde no m na do r . Po r l o t a nt o , hay q ue e f e c t u ar l a d i v i s i n5x+6 5 x + 6f\ ' 11\ p ar a ob t e n er y -, +\

1 - - 3 +

l uego x a +3 x - 2

\ /

( x3) dx+x a +3x+2

5x+6( x +1 ) ( x +2 )

( x +1 ) ( x +2 )f ( 5x +6 ) d x

( x +1 ) ( x +2 )1 2 0


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5x+6 A B+( x + . l ) ( x +2 ) x + 1 y 5x+6=A( x+2) +B( x+1) y s i s e hac e x=-

> = l s e v e que B - 4 y A =1 , l ue goT 5x+6x - 3) dx + ( x +1 ) ( x +2 )

; a +3 x - 21 4( ) dx =; + l x +2

I 3x + 1 n j x + 1 j +41n j x +2 J +cEri f or ma g en er a l , s i s e q ui e r e c a l c u l a r l a f f ( x)g < x > d x . d o n degr ado f ( x ) > g r a d o d e g ( x ) , ha y q u e h ac e r l a d i v i s i n f ( x )g ( x )R ( X)q( x) + , d on de q ( x ) , R( x ) s o n p ol i n omi o s y g r a do g ( x ) > g r a dog ( x ) R ( x )( q ( x ) + ) dx( x ) ; y as i a pl i c a r l o que s e c o no c e pa r a c a l c u l a rUt i l i z ando f r a c c i o ne s p ar c i a l e s , c ompr o ba r l os s i g ui e nt e sr es ul t ados

g ( x )

+c

1 2 1


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r ( 8 x 3 +7)5.

6 .

.

8 .

9 .

10.

11,

dx = 1nI x+1 +c( x+1 ) ( 2x +l ) 3| 4> a+x + l

: 3 ~1

x 4 d x

( 2 x +1 ) a 2x + ldx = 2 1n | x - 1 J +l n | x a +x +l ) +c .

A r c T a n X B= ;;+ _ Ar c T a n ( x / 2 ) +c .i ' * +5xa+4dx 1 1= I n I X I - _ I n ( x a +l ) + +c( x a +l ) a

r ( x + 2 ) d xx a ~4x+4

dx

2 ( x a +l )I n I X +c .

I n( x a - l ) ar d x i

x * - l 4x a d x

x +1X 1 +c

I n X .1x +11

2 ( x a - 1 )A r c T a n x +c .

+ A r c T a n ( x +1 ) +c .( x a +2x+2) a ( x a +2x+2)P3x*+4X3 + 1 6 X a +2 0 X+9

14.

15.

164

( x +2 ) ( x a +3 ) a3 X +3 X3 - 5 X A + X - 1

dx = I n j x +2 J +1 nI x a +3| +c; a +3

x a +x - 2x a - x +1

( X1 ) ( x a +2x +5 ) a

dx = X 3 +X + - I n

dx

X 1y + + C

1 6 x+1 1I n J x- 1 J I n I x a+2x +5| - Ar c T a n( ) - _ 5x +19 +c128 64 x a +2x+5

/I1 2 2


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r d>16. .1 1 1- Ar c T a n x + [ - _ 1 n ( x - xZ*- ' a+l ) +2 (3 ] ? 1 / 1 ^ 2

l / 2

.1A r c T a n ( 2 x ~ 3 1 ' a ) 3+ _12 " i l n ( xa +3 1 / 3 x +l ) +2 ( ) . Ar c T a n ( 2 x +3 A ' :

f x adx17. 1 + ; ( l +x 2 - 2 1 / 2 x ) ( l +x a +2 4 ' a x )2 x ' a x a x 2 x / ' a+ 2 i ' " ax( l n +2Ar c Tan ) +c8 ; a+x2 J - ' a+l l - x a18.

19.

' x 3 +x+2x*+2x a +l

x 4 d xx * + 5 x a + 4

dx = l n ( x a +l ) + + Ar c Tanx +cx a +l.1. 8+ _ A r c T a n x - - A r c T a n ( x / 2 ) +c .

4f dx20. x a - 6 x +5n

21. 2 x - i( x +3 ) a d x = 2 1 n ( 5 / 31. 4150

1x*+5

5x dx =02 +6

x +2 1 3dx = _ l n - +( x 2 ) a 2 4-8

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1. 22 I N T E B R A C I ON DE A L GU NA S F U NC I ON E S I R R A C I ON AL E SLas i n t e gr a l e s de l a f o r ma

ax + b ax + b ax + bR ( x , ( )Ff , ( , . . . , ( ) ? ) d x .c x +d c x+d c x+d ax + bDonde R e s u na f u nc i n . i r r ac i onal en l a v a r i a b l e s x , ( ) ? ' u . .cx+dy Pl , qi , P 2, q 2, - . . , Pr. , q n ; s o n n me r o s e nt e r o s po s i t i v o s ; s e p ue de

ax + br e s ol ver , s i s e ha ce l a s u i s t i t u c i n = Z m d o nd e m e s elcx+dm ni mo c o m n m l t i p l o d e q * , q a , . . . , q ,

xd;Ej empl o 1. Ca l c u l a r ( l +x ) * ' a - ( l +>Sol uc i n.mc m { 2, 3 }=6 , l u e go .l + x=z">, dx =6z dz y xd;

( z * - l ) 6z d;3 _. a;

( z * - l ) z 3 d 2z - 1 6

(1. +x )x'S!-(l + x( z B +z 4 +z +z 2 +z +l ) z 3 d z =

6 6 6 6 6 ea 2 -f 51+ ~ ** 1Z3 ) d Z ~ Z *** Z& z 7 ~~7 **"+ Z0 + Z ** C9 8 7 5 469

68( 1 + ( 1 + X )

E, " " S' +- ( 1 + X ) 1 + X ) + - ( 1 + X ) E,


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zadxz : s +l z

2 d z -- 4 s2 dz 4 4 - . 3 _: 3 +l I n | z

3 +l | +c

I n +1 I +c .

V e r i f i c a r l a s i g u a l d a d e s s i g u i e n t e s .dx 10 10= 1 n ( +c .: ( x ^' ^+x 2 ' 0 )

dx[I +vI/ I BUB ..-i/m ... x / a o , , / JL ' i \ ) rt A jL A

2 x 1 / a - 3 x 1 / 3 - 8 x 1 / 4 +6 x A / < f a +4Bx 1 a +3 1 n ( l +x 1 / 1 2 ) '171 2x i ' i =t - l

l n ( x i / * +x i / i s +2 ) - ( A r c T a n ( ) ) +cr 1x dx( = l n1 + x x1- x

u= ( 1+x

7 1 / 2u a - l

( u 4 +u 2 +l ) i / 2

7 Jl +2 u a+x 1 / 2 A r c T a n ( ) +c ;

4. ( ) 1 / 2 d x = I n1 + x( x +1 ) 1 - x )( 1 + x ) * ' + ( 1 - x

1- x+2 A r c T a n ( ) A ' a +c1 +xdx r>. > x y 2 _ o* 2 i / 2 A r c T a n ( ( x / 2 ) 1 ) +c

. . < +/ \ 1

t. 6 x 1 ' +3 x i / 3 +2 x i / 2 - 6 1 n ( l +x 1 / A ) +c .

1 . 2 3 I N T E G R A C I ON DE F U NC I ON E S H I P E R B OL I C A S .L as i n t e gr a l e s de l as f u nc i o ne s h i pe r b l i c a s , se? t r a ba j a n en

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f or ma muy a n l o ga a l as . i nt egr al es de f u nc i o ne s t r i g on o m t r i c a s ,e s por e l l o q ue s o l o n os l i mi t a r e mo s al c l c u l o de a l g n as

i nt egr a 1 e s -1. Ha l l a r S e nh xd xSo l uc i n.

e * + e -Senhxd: +c = Co s hx +c

2 . Ha l l a r Co s h xd xSo l uc i n.

ga x + pa xCos h x d x e " - e " ~+c Senhx+c

Ha l l ar - Ta n hxd:Sol uc i n.

Tanhxd; S en h xC o s h x dx~duu I n I uI +c 1 n I C o s h I +c

Ha l l a r Co t h xd x Cosh x5enhx dxduu I n I uI +c 1 nI S en hx I +c

5 . Ha l l a r S e c h x d xSo l uc i n.

S e c h a x d x = T a n h x +c y a q u e l a d e r i v a d a d e T a n hx e s i g ua l a

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Sech 2 :-; .

6. Ha l l a r C B C h2: : D:S o l u c i n .

c s c h 2xd x = - Co t h x+c

7. Ha l l a r S e c h x T a n h x d ;

Sol uc i n.S ech xTan h xd x = - S ec h x+c

8. Ha l l a r Cs c h x Co t h x d x .So l uc i n.Cs c h x C o t h x d x = - C s c h x +c

9. Ha l l a r ( a 2 +x 2 )Sol uc i n.

d x d= A r c S e n h ( x / a ) +c . Y a q ue ( a 2 +x 2 ) i ^2 dx

( Ar c S e nh ( x / a ) +c )

( a 2 +x 2 ) 1 X 2

10. Ha l l a r . dx( x 2 - a 2 )1 2 7


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Solucin.

11. Ha l l a r

Ar c Co s h( x / a ) +c

Senh^xdx

Sol uc i n.

Senh^xdx12. Ha l l a rSo l uc i n.

Sech*xdx =

( Co s h 2x - l ) d x = - S e n h2 x - _ +cSech4: : Sec h3x d :

T a n h 3 xTanhx- +c

13. Ha l l a rSo l uc i n .

e MC o s h x d x

e K C o s h x d x

e w+ e - ie" ( ) d x = - 1 x( e z - +l ) d x = - e a " + _ +c4 214. Si t i e ne una i nt e gr a l d e l a f o r ma R ( x , ( a 2 +( c x +d ) 2 ) 1 / 2 ) d x ,d on de R e s un a f unc i n r a c i o na l en l as v a r i a bl e s x , ( a 2 ( c x +d ) 2 ) 1 / 2s e p ue de r e s o l v e r s i s e h ac e l a s u s t i t u c i n c x +d =a S e nh u ; a >0 .

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15. Si s e t i e ne u na i n t e gr a l d e l a f o r ma R ( x , ( a 2- ( c ; - : +d) 2) 1 / 2) d x ,donde R e s u na f u nc i n r a c i o n al , s e p ue de r e s o l v e r s i s e h ac e l as u s t i t u c i n c x +d - a S e c h u ; a >0 .16. Si s e t i e ne u na i n t e gr a l de l a f o r ma R( x , ( ( c x +d ) 2 - a 2 ) 1 / 2 ) d x ,en a l g un as o po r t u ni d a de s s e pu ed e r e s o l v e r s i s e ha c e l as u s t i t u c i n c x +d =a Co s h u ; a >0 .17. Ha l l a rSo l uc i n.

Senhxdx .

xSenhxd; eH - e - H 1;.;( ) d X = - x e " d x

( x e w- e " ) - - ( - x e - K - e - H ) +c = x( e + e - e - e -i - - +c: C o s hx - Se nhx +c

1 . 2 4 E J E R C I C I OS .Ver i f i c ar l as s i gu i e nt e s i gua l dades ,1. Cos h^x Sen h 3 xS e n h 3 x C o 5 h 2 x d x = +c

Co t h 3xCot h^Sxdx

eh 3xdx C o s h3 x

- +c .

Coshx+c

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d x = Ar c Co s h ( x / 5 ) +c( x a + dx = Ar c Sen hx- +c

dx Ar c T a n h( 3 x / 2 )4 - 9 x a

xCoshxdx = xSenhx - Coshx+c

( x a - 9 ) 1 / 2 d x = - - Ar c Co s h ( x / 3 ) +c .d X 1= - _ Ar c Co t h ( x +( 3 / 2 ) ) +c .4 x a +12x+5 4

dx Ar c Se nh( ( x - 1 ) / 4 ) +c .( x a 2x +1 7)C o s h X3x a S e n h x 3 d x = +c

e" Ar c T a n( e M/ 4 )dx = +c ; si . e "


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16.Son h' 1 1 a x Cos h a x ( n - 1 )Senhna ; ; d ; Xx , . . . , xr. J u na p ar t i c i n d e [ a , b ] d on de c ad as ubi nt er ba l o [ x k - i , x k ] t e nga l a m s ma l ongi t ud ( h =( b - a ) / n ) .Se un i r n 1 o s pun t o s ( x k _ i , f ( x k - i ) ) y ( x * , f (x * ) ) med i an t es e gme nt o s d e r e c t a ( c o mo s e o b s e r v a en l a f i g ur a a n t e r i o r ) ;par a f o r ma r n t r a pe c i o s , c on r e a del k - s i mo t r a pe c i o

hAk = - [ f ( x _ J +f ( x k ) ] .OCon m s p r e c i s i n, s e di r i a r e a d i r i g i d a, p ue s t o qu e Ak. s e r nagat i v a pa r a un i n t e r v a l o d on de f s ea n eg at i v a . L a i n t e g r a lb

f ( x ) d x e s a p r o x i ma d a me n t e i gua l a. A A +A a +. . . . +Ar; es dec i ra

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b h h hf ( x) dx ; ; _ [ f ( x, ) +f ( x * ) ] +- [ f ( x ^+f ( : = a = 1 . 0 0 ; f ( x ) = ( 1 . 0 0 ) 4 = 1 . 0000.K l = a +h = 1 + h = 1 . 2 5; f (xj , ) = ( 1 . 2 5) * 2 . 4 4.x a = a+2h = l +2h = 1 . 50 ; f ( x 2 ) = ( 1 . 5 0) * ~ 5 . 0 6.x 3 = a+3h = l +3h = 1 . 75 ; f ( x 3 ) = ( 1. 75) * - 9 . 37.; . u = a+4h = 1+4h = 2 ; f ( x* ) = ( 2 . 00) * ss 16 . 00 .

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a+5ha+6ha + 7ha+8h

l + 5 h =l +6h =l +7h =l +8h =

2. 25; f ( x a)2. 50 ; f ( x * )2 . 75; f ( x y)3; f ( x a )

( 2. 25) * 2 5. 6 2.( 2. 50 ) * ~ 3 9. 0 6.( 2. 75) * 5 7. 19.3 * ~ 8 1 . As i que

0 . 25 [ 1 . 0 0 +2 . 4 4 +5 . 0 6 +9 . 3 7 +1 6 +2 5 . 6 2 +3 9 . 0 6 +5 7 . 1 9 +8 1 ]

48. 94 y e s t e s e pue de c o mpa r a r c o n e l v a l o r e x a c t o de 1,

i nt egr al 'dx 242 4 8. 4 0,

E j e mp l o 2 . Ap r o x i ma r ( 4+; - ! 3) 1/ =dx me d i a n t e l a r e g l a d e l o s0

t r a pe c i o s s i n =4 .So l uc i n.2 - 0 2 1 b - a 1Cada s u b i n t e r v a l o t i e ne l o ngi t ud = = _ = _ , l ue go = h = 4 4 2 ' n 2

P = 0, ( 1/ 2) 1, ( 3, 2) , 2] - =[ x, Xi , x 2 , x 3 , x } e s l a pa r t i c i n d e[ 0 , 2 ] ,

e n t o nc e s 1( 4 +x 3 ) 1 / z d x - [ f ( 0 ) +2 f ( 1 / 2 ) +2 f ( 1 ) +2 f ( 3 / 2 ) +f ( 2 ) ]40

[ 2+4. 06 +4. 4+5. 43 +3 . 46 ] = 4 . 85 , d on de f ( x ) =( 4 + x 3 ) x .1 . 2 7 . 2 R E G L A DE L OS R E C T A N GU L OS .

1 3 9


28/60

Sea y =f ( x ) u na f u nc i n c o nt i n ua en el i n t e r v a l o c e r r a do [ a, b3 y[ a, b] d i v i d i d o en n s u bi n t e r v a l o s de i gua l l o ngi t ud , y s e a x A elpunt o me d i o de l i s i mo s u b i n t e r v a l o , e n t o n c e ?

baf ( x ) d 5 Cf ( Xa ) +f ( x i ) +. . . +f ( Xn) 3 y l a a pr o xi ma c i n e s bi en

buena s i n e s s u f i c i e n t e me n t e g r a n de ,

E j e mp l o 1 . Ap r o x i ma rr e ct ngul o s .So l uc i n.

xdx s i n=4 por e l mt odo de l os

-f(x) =x

b- a 2+2Cada s u b i n t e r v a l o t i e ne l o ngi t ud = =1 , u na p ar t i c i nn2 - 1 - 1 - 0 1de [ - 2 , 23 e s P ={ - 2 , - 1 , 0 , 1 , 2} y as i x 0 =

0+1 1 1 + 2x 2 = - , A 3 , l ue goo o

3 1 1 3x dx a l [ f ( - - ) +f ( - - ) +f ( ~) +f ( - ) 3-1 1 3_ + _ + _O n 0 ,

1 . 2 7 . 3 R E B L A DE S I MP S ON .Ot r o m t o do pa r a a pr o x i ma r el v al o r de u na i n t e gr a l d ef i n i d a l o

1 4 0


29/60

pr opor c i ona l a r eg l a de S i mps o n , per o a n t es de en t r a r ende t a l l e s , s e d emo s t r a r un t e or e ma d e g r a n u t i l i d ad .TEOREMA Sean (;;, y ) , ( x x , y x ) ( x a , yz ) t r es punt os no c o l i n ea l e sen l a pa r bo l a que t i ene por ec ua c i n y =Ax : i +Bx+C, do nd e ya, , y*, yy= son mayor es o i gua l que cer o , Xi =; : 0+h, x a=x e, +2h , en t o nc es l amedi da del r e a d e l a r e gi n a c ot a da por l a p ar bo l a , el e j e

h: ; , l a s r e c t a s x = x, a, Y x =x a es t a dada por ( ya+4y i +y 2) ( Ver f i gur as i gui ent e)

De mo s t r a c i n .Como (XB> ya) , ( Xi , ya. ) , ( x a , y 3 ) s o n t r e s p un t o s d e l a p ar bo l a ,s us c o o r d e na d as s a t i s f a c e n l a e c ua c i n d e l a p ar bo l a . As i cuando s e s us t i t uy e x A por x+h, x a po r x a +2 h s e t i e ne :Y ia A H B + B X O+CYx = A ( X c a +h ) + B ( x+h) +C = A ( x a +2hx B +h a ) +B ( x+h) +Cy a = A ( x


30/60

1 B( Ax a +Bx+C) dx = - Ax 3 + - x 2 +CxT OXe+2h

- h [ A ( 6 xe : 2+12hxsB+8h3t ) +B( 6xa+6h) +6C] ( e j e r c i c i o )

h[ y ta'1"4y i +y a ] .El m t o d o d e S i mp s o n c o ns i s t e en d i v i d i r el i n t e r v a l o [ a, t a] ,en 2n s u b i n t e r v a l o s d e i gual l o ngi t ud y s e a pl i c a el t e or e maant er i or ; e s d ec i r :Sea y =f ( x ) u na f unc i n c o nt i n ua en el i n t e r v a l o c e r r a do [ a , b ]y el i n t e r v a l o e s d i v i d i d o en n s u bi n t e r v a l o s de i gual l o ng i t ud ,es de c i r h = x =( b - a ) / n , n e nt e r o par , e nt o nc e sb h hf ( x ) d x C f ( Xa) +4f ( x j . ) +f ( Xa) ] +- Cf ( x 2 ) +4f ( x 3 ) +f ( x ) ] +

. . + - Cf ( x _ a ) +4 f ( X n - J + f ( Xn) ] =b- a 1[ f ( Xa> ) +4 f ( Xx ) +2f ( x a ) +4f ( x 3 ) + . . . +2f ( Xna ) +4f ( x-i ) +f ( x ) ]nn s u f i c i e nt e me nt e g r a n de .

1 4 2


31/60

Ej e mpl o i . Ap r o x i ma r dx - , p ar a n=4 , p or S i mp s o n .

So l uc i n.3 - 1 ? '>oJ. , ai - / i . , Xa . , A3d/ x. , X4-_> 3 as i

dx2* 3-C f ( X , ) + 4f ( x i ) +2f ( X A ) +4f ( x 3 ) + f ( ) 3

- Cf ( l ) +4f ( 3 / 2 ) +2 f ( 2 ) +4f ( 5 / 2 ) +f ( 3 ) 361 2 1 2 1 1 15+40+15+24+5 1 99 11- C1+4 . - +2 . - +4 . - +- 3 - - [ 3= - C 3 =6 3 2 5 3 6 15 6 15 10

E J E R C I C I OS V AR I OS .Ver i f i c ar l as s i gu i e n t e s i gua l da de s :

C o s x dx1 ( 1+Senxdx

( 1 + x ) x * ' 2dx

( e" + lCo s 2 t d t i1+Sen2t 2

2( 1+Senx ) +c

2 Ar c i anx 1 / 2 +c

l n ( 1+e" )( l . +eK ) ' +1 +c

I n I 1 +Se n2 t I +c .

1 4 3


32/60

2Cosx( 1+S en x )S en xd x

Co s a x - 5 Co s x +4dx

H- C( 1+Senx

I n 1 - Co s x4 - Co s x +c

dxI n I e K - l I - x +c

X d + X 1 ' 3 )dx

I nj X I - 3I n I 1 + x 1 ' 3 | +c1 [ 4 x - 3 1 n ( l +e 2 " ) +l n ( 3 +e a ) ] +ce 4 +4 e 2 +3 12

8d X 2M 1 J f\

+ - + I n: + 2

+c

l n t ( x 2 +l ) 1 / 2 ] d x = X I n ( x 2 +l ) ^' ^- x +Ar c T a n x +c .S e c a x d x

( 4--Tar)=;. ; j i / sTan XA r c S e n ( - ) +c

1 n ( X- ( Xs * ) 1 / 2) d ; : = x 1 n [ x - ( x - ! ) 1 ' 2 ] + ( x 2 - l ) 1 / 2 +c

e - HAr c T a n e " K d x = x - e - " A r c T a n e " l n( l . +e=! K) +c

I n ( x +x l / 2 ) d x = x I n ( x +x 1 / 2 ) - x +x 1 / 2 - l n ( l +x i / 2 ) +c .

1 n ( x =*+x ) d x = x 1 n ( x x ) +1 n ( x +1 ) - - 2x+c

14 4


33/60

17. C o s x A ' a d x = 2;


34/60

27

28.

T C / 2

- n / 2T I / 3

-jTC/ 6T T/ 4

Se naxCos 3xdx415

Tan32xSec a2xdx = 0.

29dx

( 1- Senax )*''l n ( +2 y : 2 )

0

30. Sen 3 xCos2 x d x = 0.nITSena3xdx = TC ,

SerunxSennxd x = 0,0

3 4 ,

55.

dxx3+x

x d x( x- 2) 2dx

gj M3 G>

I n ( x2+l ) +c .

I n +c .el n +ce' e>

1 4 6


35/60

x * dx i 8= x +A r c T a nx - _ A r c T a n ( x / 2 ) +c4 3 3( x * +l ) dx 1= 1 n I X I + +c .x ( x a +l ) 2 x 2 +lf t r c T a n x3 V 2 d x = ( Ar c T a n x l / 2 ) 2 +cx l / 2 ( 1 +x )

2 X ' ^''. Md X = +c .1 n 2dx

1+e* I n ( 1+e" ) +c

S en ( l n ( x ) ) d x = S en l n x- Co s l n x+cdx

( A r c S e n x ) 3 ( l - x 2 ) 1 / 2 2 ( Ar c S e nxx 3 d x 1= _ A r c S e nx 4 +c .( l - x B ) 1 / a 4

1- x( ) x / 2 d x = A r c S e n x +( l - x 2 ) i / 2 +c .1 + xX 4 1 1 1dx = c ~ - X4*- - X 3 - - x - x - l n I 1- x1- x 4 3 2Ar c S e n x d x I ( x +1 ) i / 2 Ar c S e nx +4 ( 1 - x ) * ' 2 +c( 1 + x) 3-x - 2 x 3 +3 x58 - x+3 1dx = - x 3 +l n

% j L A T A ( x 2 - 2 x +3 ) 1 / 2

1 47


36/60

r x A' ad -;> w a /1 + X

dx( 6 +x - x a ) 1

dx

2 Ar c T a n x 1 / 2 +c2X--1A r c S e n ( ) +c5

3+5Senx 4e* * MSenbxdx

e * MC o s b x d x

I nB"

3 Ta n( x / 2 ) +l3 +T a n( x / 2 ) +c

a a +b a ( a S en bx - b Co s b x ) +c .e< ( bSenbx +aCos bx ) +ca 2 +b 2

r ( x + .1 ) d x( x a +2 x - 4 ) ( x

a +2 x - 4 ) x 's+c

x +1 x+1( 3 - 2 x - x a ) * ' a dx = ( ) ( 3 - 2 x - x a ) 1 ' a +2 A r c S e n ( ) +cdx

l +Co s 3 xl - Co s 3x

3Sen3xx 4( x a 8x) 1 / a d x

( 6 x x a ) 1 / 2 d x

dx( 2 0 +8 x - x a )

2x+3

+c

( x a 8x ) 1 / 2- 8 1 r i I x4+ ( x a - 8 x ) x y S i i +c

( 6 x - x z ) 1 / 2 +- Ar c Se n ( ) +cX4Ar c Se n ( ) +c

1 139 x a - 1 2 x +8 d x - 1 n | 9 x

a 1 2 x+8I + Ar c Tan ( ) +c9 18 2

1 4 8


37/60

60,

61

( a a - x a ) 1 / a d x = - ( a=- ; ; a) i ' ' =2+ Ar c Se n ( x / a ) +c .

63.

64.

65.

66.

67.

68.

69,

70.

71.

( a2- ; - : 2)

x 2 d x( a =- x 2 ) 1 / 2

d>

dx = ( a 2 - x a ) 1 / 2 - a l n a +( a 2 - x 2 ) i / : +ca- 5

x 2 ( a 2 - x a ) 1 / a( a a - x a ) s / a d x

( a - x Ar c S e n( x / a ) +c .

( a 2 - x a ) i / a +ca- 'x- ( 2x a- - 5 a; z ) ( a 2 - x 2 ) - Ar c Sen ( _ ) +cS 8 a

a"x 2 ( a a - x a ) l / 2 d x = - ( 2 x 2 - a 2 ) ( a 2 - x a ) ^2 + A r c S e n ( _ ) +c .8 8 a" ( a a - x 2 ) 1 / a d x

d;( a 2 - x 2 ) 1 / 2 - A r c S e n( - ) +ca

x ( a 2 - x a ) 1 / :dx

( a a - x 2 ) 3 / ax a d x

l n +c

+ ca 2 ( a a - x 2 ) 1 / :

( a Ar c S en ( x / a ) +csa _ s a ) ( a=- x=* )1 l na a - x a 2a x+ax - a +c .

dx = ( a 2 +x 2 ) 1 / 2 - a l n a +( a 2 +x 2 ) x y +c

1 4 9


38/60

72.

73.

74,

75.

76.

77,

78.

79.

80,

8.1.

82.

83,

a*( a a +x 2 ) 1 / 2 d x = - ( a a +x a ) I n I x + ( a a + x a + (dx

( a 2 +x a ) 1 / adx

I n I x +( a a +x 3 ) 1 / a | +c

+c .( a a +x 2 ) " ' 2 a =( a 2 +x 2 )( a +X )*'= ( a 2 +x 2 ) 1 / 2dx + l n l x +( a 2 +x 2 ) 1 / a | +c

1 d X Ar c T an ( x / a ) +c( a 2 +x 2 ) aa*( ;


39/60

84.

85.

86.

87.

88.

89.

90.

91

92.

93.

94.

95.'

Cos - axSenax n - lCosna:-:dxna nCo s ( a+b ) x Cos ( a- b ) xSenaxCosbxdx

SenaxSenbxdx2( a+b)

Sen( a- b) x: ( a~b)

Sen( a- b) xCosaxCosbxdx = +2( a- b)

2( a- b)Sen( a+b) x

2( a+b)Sen( a+b) x

+c ;

+c ; aaj =b:

+c ;: ( a+b) a2 ^b2

Sen" axCosaxd x Senn +1ax

a( n+1) +c ; n =- l .

c ; enna xCos ma x d x

S e n" a xCos ma ; ; d ; ;

Sen" - ' 1axCo5' n, 1ax n- l+a( n+m) n+m

Se n" 1 a x Cos"' " 1 a x m~ 1+a( m+n) m+n

Senn- 2axCos maxd:

SennaxCo5m- aaxdx

xnSenaxdx =

x" Cosaxdx

xnCosax n+ _a a

x" Senax n

Tar i naxd!

Cot naxdx

Tan"" 1axa( n- l )

- Cot " - 1 axa( n- l )

Cosaxdx

Senaxdx

Tann - 2axdx n==..

Cot r , aaxdx ; n l .S e c " 2axTanax n~2Sec"axdx = +a( n- l ) n- l .

c ; ec n- 2axdx; nJ =

1 5 1


40/60

CsC' axdx

' Sen- - axd;

x nCo s - 1axdx

; - : "f t rcTana; ; dx

Cs c n" 2axCot ax n - 2 +

a ( n - 1 ) n - 1xn+' i Sen~1ax a

n + 1 n + 1a

Cs c n _ 2axdx ; nl., n JL H > 'A U A

Cos - 1ax +n + 1 n + 1xn+1f i r c Tanax a

( l - a^x3 5 ) =x n +1 dx

( l - a 2x 2 ) 1 / :: n +1dx

n - 1

n i - 1

i nb" Kn + 1n

n+1 l +xaa= ; n - 1 .

; r' b * " d ;

" I n a x d

Sen hr , axd x

/ r X i--, aa 1 n b a 1 n b ,x n + 1 1 n a X

n+1 ( n+1)Senh n _ l axCos ha x n- 1

b" Mdx ; b>0, bz|=

+ C ; nr | r - l .

n a nCoshn- 1axSenhax n - 1Cosh"axdx = +

x" Senhaxd x

x"Coshaxdx

Tanh" axdx

Co t h"axdx

nax " C o s h a x n

a a Jy," n Senhax - a a ,

n

Senhn _ 3axdx ; n

Coshn - 2axdx ; n =0

Cosaxdx.

x" - 1Senhaxdx.

Tanhn- 1ax +( n - 1 ) aGot h"" 4ax

( n - 1 ) a

Tanhn - 2ax dx ; n l. .

Co t hn - a axdx; ' n i l

1 5 2


41/60

Se c h^- â x T a nha x n - 2Bec hnaKd: <

Cs c hâx dx = -

x " ( a +b x ) 1 / 2 d ;dx

x n ( a +b x ) x y s td x

( n -1 ) a r i -1 ,Cs c h^- â x C o t ha x n- ( n - 1 ) a2 x " ( a +b x )

n - 1

SBchn~ax d!


42/60

n/ 4119. dx 1 3 # 2

1 ' a= _ i n (2+Tanx 5n) +4 100

n/ 3120. 1 - S en x = 3

A ' a +l .0n / 2

121.dx - n

3+Cos2> 80

122. = I n ( 21 ' a- - l ) .( x a +2x+2)5

123. A pr o x i ma r ( 3 5 +x ) 1 / 2 d x . a ) me di a nt e t r a pe c i o s n =4 . b )

F r mul a de S i mps on n=4 .

124. A p r o x i ma r

12 5. A pr o x i ma r

n

01

0

Senxdx por l a f r mu l a de S i mps o n n=6 .

( l +x 3 ) 1 / 2 d x a pl i c a n do a ) F r mu l a de l os

t r a p ec i o s ; n =5 . b ) d e S i mp s o n ; n =4 .1 . 2 8 I N T E GR A L E S I MP R OP I A S .

1 5 4


43/60

En el e s t ud i o de l a i nt egr a l de f i n i da f ( x ) d x , s e haa

s o b r e nt e nd i d o, h as t a a ho r a q ue :1. L o s l i mi t e s de i n t e g r a c i n s on n me r o s f i n i t o s .2. L a f unc i n f ( x ) e s c o nt i n ua en el i n t e r v a l o [ a , b] . Si f esdi s c o nt i n ua d eb e s e r a c ot a da en e s t e i n t e r v a l o .Gu an do s e e l i mi n a u na d e e s t a s d os c o n d i c i o n e s , s e d i c e q ue l ai nt egr a l r e s ul t a nt e e s una i nt egr a l i mpr o pi a ; en o t r a s pa l a br a s

bl a i nt egr a l f ( x ) d x ; s e d i c e i mpr o pi a s i

.1. *f=~o b~+


44/60

bi nt egr al i mp r o p i a de s e g un da e s p ec i e . Y s i l a i nt e gr a l f(x)ds

apr es ent a l as d os c o n di c i o ne s a nt e r i o r e s , s e l l a mar i nt e gr a l

/ - ^i mpr opi a d e t e r c e r a e s p e c i e .Ej e mp l o s .00 0 d) 00

dx 10

C o s x a d x ; S e nx d x ; e ~Kd x0 a> 0de p r i me r a e s p e c i e .

10L as i n t e gr a l e s

- 1de s e g u n d a e s p e c i e .

x d x ; s on i n t e gr a l e s i mpr o pi a s

16dx( x - 1) ( x - 5) x- 7 ; s on i n t e g r a l e s

Y 1 as i n t e g r a 1 e s * dx dx( x ~1) ( x - 2 ) ( x - l ) 1 ' 2 ; S o n0i n t egr a l e s de t e r c e r a e s pe c i e .Se h ar un e s t u d i o d e t a l l a d o d e c a d a u na d e e l l a s .1 . 2 8 . 1 I NT E G R A L E S I MP R OP I A S DE P R I ME R A E S P E C I E .Sea f ( x ) a c ot a da e i nt e gr a bl e en un i n t e r v al o c e r r a do [ a, b ] . S e

0D bf ( x ) d x .ef i ne f ( x ) d x = 1 i mb>a

1 5 6


45/60

La i nt e gr a l f ( x ) dx s e d i c e c on ve r g ent e ; s i l i mb>a> f ( x ) dx e x i s t e ;a

en c as o c o nt r a r i o l a f ( x ) d x s e d i c e d i v er ge nt e ,b

Cuando l i mb- >a> f ( x ) dx = Ae R; s e d i c e q ue el v a l o r d e l a i n t e gr a iaf ( x ) d x =A. En f o r ma a n l o ga s e d ef i n e l a f ( x ) d x= l i ma- >- oo f ( x ) d x .

E j e mp l o 1. Mo s t r a r qu eval or ,

e ~K d x e s c o n ve r g en t e y h al l a r s u0

So l uc i n.

e~" dx 1 i mb>oo e ~K d x l i m - e - "b>oo0

= l i m ( l - e_ f c >)0 b- >oo .1. Lu ego

l a i nt egr a i e ~K d x c o n v e r g e y s u v a l o r e s 10

E j e mp l o 2. Mo s t r a r q ue l a i n t e g r a iha l l ar s u v al o r

d xe " +1 e s c o n v e r g e nt e y0

1 5 7


46/60

Solucin.

b b bd x= l i me M+1 b- >

dxe K +l

0

l kmB->o [ x - l n ( e M + l ) ]

1 i mb- >


47/60

Solucin.

0e Md x = 1 i ma > e d = 1 i m e a- >- oo

e " dx e s c o n v e r g e nt e y s u v a l o r e s 1

E j e mp l o 5 . Mo t r a r q u edx

e s c o n v e r g e n t e p ar a p >l yad i v e r g en t e p ar a pi 1, s i a >0 .So l uc i n.dx = 1 i mx p b- > ,

b b' dx = 1 i mK p b> 00y (3 + 1~Rd x -- l i mb~ >oo 1 - pa

- a1- - s i p>la

1 - P+ s i p c o

' dx 1 i. m 1 n xb >oo

b i - p _ a i - p1 i ma b- >oo i ~p

dxa

= l i m ( l n b- l n a) = +


48/60

De f i n i c i n . Si f ( x ) es c ont i nua par a t odo x , l a i nt egr a l

f ( x ) d x s e d e f i n e po r :

c u a l q ui e r n me r o r e a l

f ( x ) d x f ( x ) d x + f ( x ) dx , s i e ndo aa

Si a mb as i n t e gr a l e s f ( x ) d x y f ( x ) dx c o nv e r g e n e nt o nc e s l aa

i nt egr al f ( x ) dx c o nv er g e y s i c ua l qu i e r a d e l as i n t e gr a l e s

f ( x ) d x y f ( x ) d x d i v e r g e en t o nc e s f ( x ) dx d i v er ge ; s i

f ( x ) d x = A y f ( x ) dx = B; en t onc es f ( x ) d x = A+B.

E j e mp l o 1. Mo t r a r qu eODf dxk 2 + 1

e s c onv er gent e? y h al l a r s u v al o r ,

So l uc i n .0dx

xa+. l1 d;x 3 +l

0dx:+ l 1 i. ma- > ..dx

; 2+l0 a 0

1 6 0


49/60

l i m Ar c Tan:a- > oo0 + l i m Ar c Tan:a b>oo = l i m ( 0 - Ar c Ta na ) +1i m Ar c T a nb0 a>oo b>o o

TI( TI / 2 ) + = rt ; l uego dx e s c o n v e r g e nt e y s u v a l o r e s TI.; 2+l

Ej e mpl o 2 . Mo s t r a r q ue ; dx e s d i v e r g en t e ,

0So l uc i n. xd: x d x + x dx ; Co mo l a i n t e gr a l xdx es

0 0

di v er g en t e y a q ue l i mb>a> x d x = l i mb>o0b b 2= l i m = + o o ; s e0 b- > 2

puede c o n c l u i r q ue x dx d i v e r g e .

bNo t a : x d x = 1 i mb>oo , xdx = 0 e s i n c or r e c t a , p ue s l a d ef i n i c i n

r equi er e de l a e va l ua c i n de 2 l i m t es d i f e r e nt e s .Ob s e r v a c i n 1 .

L as i n t e gr a l e s de l t i po f ( x ) d x p ue de n r e d uc i r s e n a i n t e gr a l e s

1 6 1


50/60

i mp r o pi a s d e l a f o r ma

u=- x. E n e f e c t o :

h ( x ) d x , si . s e ha c e el c a mb i o d e v a r i a b l e sa

af ( x ) d x = l i ma - > f ( x ) d x = l i ma- >a> f ( - u ) du = f ( - u ) d u .b

0E j e mpl o 1. Mo s t r a r qu e l a dxx 2 +l e s c o n ve r g en t e ,

So l uc i n.

Sea u =- x ; d u =- d x y as i> : < '

0 0r dx = l i ma +1 a- >oo J

0dx = 1 i nx 2 +l a- >


51/60

l i ma-> a>

e ~u d u - l i ma> , e _ u d u e _ u d u e - H d x q u e esa 1

c o nv e r g e nt e ; l u eg o e * d x e s c o n v e r g e nt e ,

Db s e r v a c i n 2 . S i f ( x ) e s c o nt i n u a en [ a , b) , p e r o n o a c ot a da enb

[ a , b) ; e n t o nc e s l a i nt e gr a l i mpr o pi a de s e g un da e s p e c i e f ( x ) d x

puede r e du c i r s e a una i nt e gr a l i mp r o p i a d e p r i me r a e s p e c i e ,medi ant e el c a mb i o de v a r i a bl e u =l / ( b - x ) .En e f e c t o ;b c 1 / ( b - c )

f ( x ) d x = 1 i. mc ~>b ~ ,

f ( x ) d x = l i mc - >b "

1 duf ( b - - ) u u 2a 1 / ( b - a )

i duf ( b~ ) . Si f ( x ) e s c o n t i n u a en ( a , b ] , p e r o no a c ot a da enU Ll"21/ ( b - a ) b

( a , b] , l a i n t e gr a l i mpr o p i a f ( x ) d x s e p ue de r e du c i r a u naa

i nt egr a l i mp r o p i a de p r i me r a e s p ec i e h ac i e nd o u ~l / ( x - a ) : enb b 1 / ( b - a )

e f e c t o f ( x ) d x = 1 i mc >a" f ( x ) d x - 1 i mc - >a H1 duf ( a + - ) U LL 21 / ( c ~a )

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l / ( c - a ) oc.1 dul im f ( a +) =cr>a' +' LI u 21 duf ( a+ _ ) u u 21 / ( b - a ) 1 ( / b - a )

Como t o d o t i p o d e i n t e g r a l i mp r o p i a p ue d e , me d i a n t e un c a mb i o d ev ar i abl e a de c u a do , t r a ns f o r ma r s e en u na i nt e gr a l i mp r o pi a depr i mer a e s p e c i e , s e e nu nc i a r n y s e p r o b ar n t o do s l o sr e s ul t a do s p ar a e s t e c a s o .T EORE MA. S i f ( x ) y g ( x ) ei j t n a c o t a d a s en [ a, +oo) y f ( x ) d x y

ag( x ) d x c o n ve r g e n a mb as , e n t o n c e s i ) ( f ( x ) g ( x ) ) d x c o n v e r g e y

( f ( x ) g( x ) ) dx f ( x ) d x g ( x ) d xa a

i i ) c f ( x ) dx c on ve r g e y c f ( x ) d x = c ( x ) d x .a a a

De mo s t r a c i nComo par a t odo be[ a , +) ( f ( x ) g( x ) ) d: f ( x ) d x g ( x ) d x ; s e

a a bt i ene q u e l i mb>oo ( f ( x ) g( x ) ) d x = 1 i mb>oo f ( x ) d x l i mb> 00 g ( x ) d x =a a a

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f ( x ) dx g ( x ) dx as i ( f ( x) g ( x ) ) d x c o n v e r g e y f ( x ) g( x ) ) d xa a

b bf ( x ) d x g ( x ) d x - Ade ms c omo 1i mb- > c f ( x ) d x = 1 i m cb- >a> f ( x ) d;a a

c l i mb- > f ( x ) d x = c ( x ) dx ; s e t i e ne que c f ( x ) d x c o nv er g e ya

c f ( x ) d x = c ( x ) d x

E j e mp l o 2 . Se s a be que d x y ' dx s o n c or v er ge nt . e s ,e nt o nc e s

i )

i i )

1 1 + ) dx e s c onv e r g ent e y 1 .1( + ) d:

- ) d x e s c o n v e r g en t e yW3 ) d;

' d x ' dx1 1' dx (Df d:

.1 .1

' 5i. i i ) d x e s c o n v e r g e n t e y ' 5dx

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NOTA. La i n t egr a l d xx ( x + 1 ) e s c o n ve r q en t e , s i n e mba r go.1.

d-;( x + 1 )

.1 1X+ .l ) d x

1 dx * dx - ; pues l ax +1dx y

dx: + .1 s o r d i v r q e n t e s ; p a r a d a r s e 1 a i. g 11 a 1 d a d d e t) e n ser " a mb t s1

i nt e gr al es c o n v er g en t e s ; c o mo l o d i c e el t e or e ma .La i n t e gr a c i n po r p ar t e s e s a me nu do t i l e n l a e v al u ac i n d e1 a s i n t e g r a 1 e s i mp r o p i a s . S i f ( x ) y g ( x ) t i e n e n d e r i v a d ac o n t i n u a s e n [a , - ' > ; e n t o n c e s , p a r a t o d o be II a , +)

b bf ( x ) g ' ( x ) d x = f ( x ) g ( x ) ' ( x ) g ( x ) d x. S. i s e s a be qu e d os

d e 1 o s t r e s 1 i. m.i t e s 1 i mb> u f ( x ) g ' ( x ) d x ; 1 i mb- >00 f ' ( x ) g ( x ) d x ya1 i m L'f ( b ) g ( b ) - f ( a ) g ( a ) ] e x i s t en , e n t o n c e s el t e r c e r o t a mb .ib- >

ex i s t e y f ( x ) g ( : ) d x = 1 i m f ( x ) g ( x )b > >af ' ( x ) g( x ) d x

a

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03E j e mp l o 1. Ca l c u l a r x e ~Kdx

0So l uc i n .

Sea f ( x ) =x ; f ' (;:) =dx ; g ' (;:) =e~Hdx ; g( x ) =- e - >

. x e ~K d x 1 i mb" > x e0 0

e~" d; l i m e"b- >0

Ej e mp l o 2 . Ca l c u l a r e""'" Cosxdx0

Se sabe que e ~K S e nx - e " K C o s xe~"Cosxdx0

, l uego0

e- >< ( Senx- Cosx )e " KCo s x d x = l i mb>oo0l i m e- b( Se nb- Co s b ) +' = v,b>

En a l g un as o p or t u ni d a de s s e n ec e s i t a s a b er s i un a i n t e gr a l d ad ac o n v e r g e o n o , s i n i mp o r t a r n o s el v a l o r a d o nd e c o n v e r g e y e spor e s t o q ue a ho r a n os d ed i c a r e mo s a mi r a r a l g un os c r i t e r i o s d ec o r v e r g e n c i a .1 . C r i t e r i o d e c o mp a r a c i n .Si f ( x ) y g ( x) s o n c o n t i n ua s en II a, +) y 0 i f ( x ) g ( x ) par a t odox [ a , +) y s i :

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i ) S. i g ( x ) d x c o n ve r g e ; e n t o n c e s f ( x ) d x c on ve r g e,a

i i ) Si f ( x ) dx d i v er ge ; e nt o nc e s g( x ) d x d i v er g e.a

De mo s t r a c i n . bi ) Sea F ( b ) f ( x ) dx y 6 ( b ) g( x ) dx; c o mo ( x ) y g ( x ) s o n

a ac o nt i n ua s y f ( x ) 0 y g ( x ) >0 ; F y G s o n c r e c i e n t e s y pa r a t o do

b [ a , +) s e t i e n e 0 1F ( b ) G ( b ). g ( x ) d x ; a s i F e s u na f u nc i n

mo n t o na ac o t a da en t o n c es l i r n F e x i s t e y l i m F =b- - > b- >

Su pCF ( x ) / x [ a , +) 1 y as i l i m F ( b ) = l i mb- > b- >tex i s t e .f ( x ) d x f ( x ) d ;

a a

i i ) Si g ( x ) d x c o n v e r g e , e n t o n c e s f ( x ) dx c o n v e r g e ; e s t oa a

c ont r a di c e l a h i p t e s i s ,

E j e m p l o 1 . Most , r a r qu e1 dxX- - +1 es c on ve r q en t e .

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Solucin.

i .1Se s abe que : c on x l ; y c omox 38+ 1 x 2

dx e s c o nv er g en t e , s e

t i e ne q ue ' dxx a +l e s c o n v e r g e n t e ya qu edxx 2 +l

' dx

E j e mp l o 2 . Mo s t r a r qu eSo l uc i n,

dx e s c o nv e r q en t e0

' d;>; = + ! X=* +1

dxx a +l l a

d xx 2 +l e s c o n v e r g e nt e ; p ue s

dx e s0 0 0

un nu mer o y' dxx 2 +i

dxy as i e s c o nv er g en t e , l uego+J.

dx e s c o n v er g e nt ex 2 +l

E j e mp l o 3 . Mo s t r a r q ue 1 n xx+ t d x e s c o n v e r g e nt e ,

So l uc i n .

I ri x i x y as i00

r 1 n Xx +l dx 1

x d xx s +l

" X d X d X q u e e s c o n v e r g e n t e y

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a s i ' 1 n xd e s c o n v e r g e n t e .x+l

E j empl o 4 . Mos t r a r que x=dxx M+ S e n 2 x + l n 2 x +2 e s c o n v er g en t e ,

Sol uc i n.

y as iS z +Senax +1 n a x+2 x33 x i B; adx

x 2 0 +Sen 2 x +l n 2 x +: es1

c o n ve r g en t e , y a q u e -d : c o n v e r g e y xa d x dx

x 2 0 + Se n 2 x +l n 2 x +2

E j e mp l o 5. Mo s t r a r q u eSo l uc i n,

dxM- e" +1 n 2 ( x + 1 ) +x* e s c o n v er g e nt e ,0

dx2+e" + ln-- ( x+1 ) + x**

dx q u e e s c o n v e r g e n t e y a s i13 0

d x!+e" +1 n a ( x + 1 ) +x4 e s c o nv e r g en t e .

E j e mp l o 6 . Mo s t r a r q ue e~ " d x es con ve r gen t e ,0

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Solucin.

e - x - d x = e~""dx + e ~K " d x + e ~Mdx y as i dx es0 0 0

c o nv e r ge nt e y a que e - " * dx y e " M d x c o n v e r g e n .0

E j e mp l o 7 . Mo s t r a r q ue r d xl nx e s d i v er g en t e .So l uc i n.

(D a>1 l nx l x; - yl n A Adxl nx

r d x ; Como e s d i v er g en t e , s e

c o n c l u y e q u ex

1 n x e s d i v er g en t e .

E j empl o 8 . Mos t . r * r que x 2 ~l dx e s d i v e r g en t e ,So l uc i n.

x a - l x a - l( x A +16)

e s d i v e r g en t e ; p ue s pa r a x> ( x 6 +16) 1 / a00r dxcomo d i v er g e e nt o nc e s x a - l( x ^+l ) d i v er ge ,

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2. C r i t e r i o .Si f ( x ) es c ont i nua par a t odo en [ a , +m) y l i m f ( x ) e x i s t e ,x - > a>ent onc es l i m f ( x ) =0 e s una c ond i c i n ne c es a r i a par a l a

c o nv e r g e nc i a de

De mo s t r a c i n .

f ( x ) d :a

Si l i m f ( x ) = L=ya. Si L >0 ; e n t o n c e s e x i s t e un n me r o Ni a t alque f ( x ) >L / 2 p a r a t o do x >N. P a r a t o do Xi , b c o n b >x x >N.b b L Lf ( x ) d x > - dx = _ ( b- Xj L ) .

L ( b - a )Como l i m = a>b -> 2

f ( x ) d x di ver c j e. Ah or a

f ( x ) d x + f ( x ) d x f ( x ) d x ; d e mo do q ue l i mb- > f ( x ) d x = +*>;a

i mpl i c a q ue l i mb- - > f ( x ) d x = +

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Calculo Integral 03 - Bernardo Acevedo Frias

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