Problemas. Páginas 272, 273, 274 . “Integración por Partes”

Para esta clase de problemas se aplica la siguiente fórmula:∫ u . dv = u.v - ∫ v . du . 1.- ∫ x sen x dx

u = x dv = sen x dxPara obtener “v” , se integra dv = senx ,en ambos miembros.De igual modo se efectua en todos los problemas.u = x ∫ dv = ∫ sen x dxdu = dx v = - cos xx(- cos x) - ∫ - cos x . dx = - x . cos x + ∫ cos x . dx =

- x .cos x + sen x = sen x - x cos x + c .

2.- ∫ ln x dx = x (ln x - 1) + c .u = ln x dv = dxdu = 1 dx v = x

x

ln x . x - ∫ x . 1 dx = ln x . x - ∫ dx = x ln x - x = x(ln x - 1) + c . x

3.- ∫ x sen x dx = 4 sen x - 2x cos x d + c . 2 2 2

u = x ∫ dv = 2∫ sen x .½ dx 2

du = dx v = - 2 cos x . 2

x (- 2 cos x ) - ∫ 2 .cos x . dx = - 2x cos x - 2 .2 ∫ cos x .(½) dx 2 2 2 2

- 2x cos x - 4 ∫ cos x .(½) dx = - 2x cos x - 4(- sen x ). 2 2 2 2

- 2x cos x + 4 sen x = 4 sen x - 2x cos x + c . 2 2 2 2

4.- ∫ x cos nx dx = cos nx + x sen nx + c . n2 n

u = x dv = cos nx . dxdu = dx ∫ dv = 1/n ∫ cos nx .(n) dx

du = dx v = sen nx . n

x . sen nx - ∫ sen nx . dx = x .sen nx - 1 1 ∫ sen nx .(n)dx =

n n n n n

x .sen nx - 1 - cos nx = x .sen nx + cos nx + c . =

n n2 n n2

5.- ∫ u sec2 u du = u tg u + ln cos u + c . u = u dv = sec2 u du ∫ dv = ∫ sec2 u dudu = du v = tg u

u .tg u - ∫ tg u du = u .tg u - (- ln cos u) = u .tg u + ln cos u + c .

6.- ∫ v sen2 3v dv = ¼ v2 - 1/12 v sen 6v - 1/72 cos 6v u + c .

u = v dv = sen2 3v dv du = dv ∫ dv = ∫ sen2 3v dvSe aplica : ∫ sen2 u du = ½ u - ¼ sen 2u ; donde u = 3v

v = 1/3 ∫ sen2 3v .(3) dv

v =1/3 [½ 3v - ¼ sen 2(3v)] v = 1/6 3v - 1/12 sen 6v

v = ½ v - 1/12 sen 6vv (½ v - 1/12 sen 6v) - ∫ (½ v - 1/12 sen 6v) dv =

½ v2 (- v . sen 6v) - ½∫ v dv + 1/12 .1/6 ∫ sen 6v .(6) dv =

12

7.- ∫ y2 sen ny dy = 2 cos ny + 2 y sen ny - y 2 cos ny + c . n2 n

u = y2 dv = sen ny dy du = 2y .dy ∫ dv = ∫ sen ny dy v = (1/n)∫ sen ny .(n) dy v = (- cos ny)

n

y 2 (- cos ny ) - ∫ (- cos ny) 2ydy =

n n- y 2 cos ny + ( 2 )∫ cos ny .y dy = - y 2 cos ny + ( 2 )∫ y cos ny . dy n n n n

Integrando por partes : ∫ y cos ny .dy .

u = y dv = cos ny dy du = dy ∫ dv = ∫ cos ny dy v = (1/n)∫ cos ny .(n) dy v = (sen ny)

n

∫ y cos ny.dy = y (sen ny) - ∫ (sen ny).dy =

n n

∫ y cos ny.dy = y (sen ny) - 1 . 1 . ∫ (sen ny) .(n)dy n n n

∫ y cos ny.dy = y (sen ny) - 1 . (- cos ny) .(n)dy n n2

∫ y cos ny.dy = y (sen ny) + ( cos ny) n n2

Enlazando y sustituyendo ∫ y cos ny.dy , en la integral original:

∫ y2 sen ny dy = - y 2 cos ny + ( 2 )∫ y cos ny . dy n n

∫ y2 sen ny dy = - y 2 cos ny + ( 2 ) y (sen ny) + ( cos ny) =

n n n n2

∫ y2 sen ny dy = - y 2 cos ny + 2 y sen ny + 2 cos ny .Ordenando: n n2 n3

∫ y2 sen ny dy = 2 cos ny + 2 y sen ny - y 2 cos ny + c . n3 n2 n

8.- ∫ x ax dx = a x x - 1 + c . ln a ln2 a

u = x dv = ax dxdu = dx ∫ dv = ∫ ax dx v = a x .

ln a

x . a x - ∫ a x . dx = x . a x - 1 ∫ ax .dx = x . a x - 1 a x . ln a ln a n ln a ln a ln a ln a ln a

x . a x - a x = ax x - 1 + c . ln a ln2 a ln a ln2 a

9.- ∫ xn ln x dx = x n+1 ln x - 1 + c . n+1 n+1

∫ x n ln x dx =

u = ln x dv = xn dxdu = 1 dx ∫ dv = ∫ xn dx x v = x n+1 .

n+1ln x . x n+1 - ∫ x n+1 . 1 dx = ln x . x n+1 - 1 ∫ x n+1 dx

n+1 n+1 x n+1 n+1 x

ln x . x n+1 - 1 ∫ xn+1.x -1 dx = ln x . x n+1 - 1 ∫ xn+1-1 dx =

n+1 n+1 n+1 n+1

ln x . x n+1 - 1 ∫ xn dx = ln x . x n+1 - 1 . x n+1 =

n+1 n+1 n+1 n+1 n+1

x n+1 ln x - 1 + c . n+1 n+1

10.- ∫ arc sen x dx = x arc sen x + √1 - x2 + c .

u = arc sen x dv = dxdu = 1 dx ∫ dv = ∫ dx √1 - x2 v = x arc sen x . x - ∫ x . 1 dx = x arc sen x - ∫ x (1 - x2)-1/2 dx =

√1 - x2 Ordenando: ∫ x (1 - x2)-1/2 dx y completando el diferencial.x arc sen x - (-½)∫ (1- x2)-1/2.(-2)x dx = x arc sen x + ½ (1- x 2 ) -1/2+1

-1/2+1

x arc sen x + (1- x 2 ) 1/2 = x arc sen x + (1- x2)1/2 =

2(1/2)

x arc sen x + √1- x2 + c .

11.- ∫ arc tg x dx = x arc tg x - ½ ln (1 + x2) + c .

u = arc tg x dv = dxdu = 1 dx ∫ dv = ∫ dx 1 + x2 v = x arc tg x . x - ∫ x . 1 dx .

1 + x2

arc tg x . x - ∫ x dx . Completando el diferencial.

1 + x2 v = 1 + x2 Falta (2) para completar el diferencial.Se aplica: dv = 2x dx ∫ dv = ln v + c v arc tg x . x - (½)∫ (2)x dx . Completando el diferencial y

ordenando. 1 + x2

x arc tg x - ½ ln (1 + x2) = x arc sen x - ½ ln (1 + x2) + c .

12.- ∫ arc cot y dy = y arc cot y + ½ ln (1 + y2) + c .

u = arc cot y dv = dydu = - 1 dx ∫ dv = ∫ dy 1 + y2 v = y arc cot y . y - ∫ y . - 1 dx .

1 + x2

y .arc cot y + ∫ y . dy. Completando el diferencial. 1 + y2 v = 1 + y2 Falta (2) para completar el diferencial.Se aplica: dv = 2y dy ∫ dv = ln v + c v y .arc cot y + ∫ y . dy . Completando el diferencial 1 + y2

y arc cot y + ½ ln (1 + y2) + c .

13.- ∫ arc cos 2x ax . dx = x arc cos 2x - ½ √1 - 4x2 + c .

u = arc cos 2x dv = dxdu = - 2 .dx ∫ dv = ∫ dx √1 - (2x)2 v = x

du = - 2 .dx √1 - 4x2

arc cos 2x . x - ∫ x - 2 dx .

√1 - 4x2

x arc cos 2x + ∫ 2x dx .

√1 - 4x2

x arc cos 2x + ∫ (1 - 4x2)-1/2 . 2x dx .Completando el diferencial.v = 1 - 4x2 Falta (- 4) para completar el diferencial.Se aplica: dv = - 8x dx ∫ vn dv = v n+1 + c n+1x arc cos 2x + (- ¼) ∫ (1 - 4x2)-1/2 .(- 4) 2x dx .x arc cos 2x + (- ¼)( 1 - 4x 2 ) -1/2+1 = x arc cos 2x - ¼ .( 1 - 4x 2 ) 1/2 =

-1/2+1 1/2x arc cos 2x - ( ¼ )(2)(1 - 4x2)1/2 = x arc cos 2x - ½ (1 - 4x2)1/2

x arc cos 2x - ½ √1 - 4x2 + c .

14.- ∫ arc sec y . dy = y arc sec y - ln (y + √y2 - 1 + c .

u = arc sec y dv = dydu = 1 dx ∫ dv = ∫ dy y √y2 - 1 v = y arc sec y . y - ∫ y 1 dy .

y √y2 - 1

arc sec y . y - ∫ y dy .

y √y2 - 1

arc sec y . y - ∫ y dy .

y √y2 - 1

y arc sec y - ∫ d y .

√ y2 - 1 Se aplica : ∫ d v = ln [v + √ v2 - 1 ]

√v2 - 1

y arc sec y - ln [y + √y2 - 1 ] + c .

15.- ∫ arc csc t . dt = t arc csc t + 2 ln (t + √t2 - 4 ) + c . 2 2

u = arc csc t . dv = dt 2du = - ½ dt . ∫ dv = ∫ dt ½ t √(½ t )2 - 1 v = t arc csc t . t - ∫ t - ½ dt .

2 ½ t √(½ t )2 - 1

arc csc t . t - ∫ t - ½ dt .

2 ½ t √(½ t )2 - 1

arc csc t . t + ∫ dt =

2 √¼ t2 - 1

v = ½ t Falta (½) para completar el diferencial.Se aplica: dv = ½ ∫ dv = ln [v + √(½ t )2 - 1 + c √ v2 - a2

t arc csc t + 2 ∫ ( ½ ) dt = t arc csc t + 2 ln [v + √(½ t )2 - 1 ]+ c 2 t 2 - 4 2

4 16.- ∫ x arc csc x . dx = x 2 + 1 arc tg x - x + c . 2 2

u = arc csc x . dv = dx . du = - 1 dx . ∫ dv = ∫ dx x √x 2 - 1 v = x arc csc x . x - ∫ x - 1 dt .

x √x 2 - 1

arc csc t . t - ∫ t - ½ dt .

3 ½ t √(½ t )2 - 1

arc csc t . t + ∫ dt =

2 √¼ t2 - 1

v = ½ t Falta (½) para completar el diferencial.Se aplica: dv = ½ ∫ dv = ln [v + √(½ t )2 - 1 + c √ v2 - a2

t arc csc t + 2 ∫ ( ½ ) dt = t arc csc t + 2 ln [v + √(½ t )2 - 1 ]+ c 2 t 2 - 4 2

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yy-1

y1-

y

y-1

y1-

yy-1

y1-

1-y1

y1

y1-

du

y v;dy dv . y1 sec arcu

dy y1 sec arc 31.

. c 1-xxln x1 cos arc x

dx1-x

1 x1 cos arcx .dx

1-xx1-(x) -

x1 cos arc x

22

2222

22

2212

2121212

212

2

22

2

2

2

2

2

2

2

22

y1

y1

y1

y1

y1

y1

==

==

++

+=

==

==

+=

=

+−

==

=====

===

+++

+=

∫

∫∫

∫

∫∫

∫

∫∫

+−

−

[ ][ ]

[ ]

[ ]

[ ]

[ ]

. c 2

1)-(x-1 -)1-xarcsen( -

2xsen arc x

1)-(x-1 -)1-xarcsen(21 -

2xsen arc x

2

1)-(x-1

2)1-xarcsen( -

2xsen arc x

2

1)-(x-112

)1-xarcsen( -

2xsen arc x

2

1)-(x-11)x-x2

)1-xarcsen( -

2xsen arc x

2

1)-(x-11)-(x-x2

)1-xarcsen( -

2xsen arc x

2

1)-(x-1

2)1-xarcsen()1x(

21)-(xsen arc x

- 2xsen arc x

1)-(x-1)1xarcsen()1x(21

21)-(xsen arc x

- 2xsen arc x

dx 1)-(xsen arc - 1)-(xsen arc x21 -

2xsen arc x

1)-(xsen arc vdx du

1)-(x-11

x-2x1dv x u

dx x-2x

x 21 -

2xsen arc x

dx x)-2(x

x 21 -

2xsen arc x

dx x-2x2

1(x) - 2xsen arc x

2

2

2

2

2

2

2

2

222

2

.2

)1-xarcsen(:doFactorizan

+

+

+

++

+

+−+

+−−+

==

====

=

=

=

∫

∫∫∫

∫

∫

∫

( )

( )

( )

( )

-

.

..

. --

.

-

x-1 sen x arcx vdx3xdu

sen x arcdv xu

.dx sen x arc x .34

3

23)x-1(x2

232x-1xsen x arcx dxsen x rca x4

3

x-12v

x-1 v;dx du

. dx2)x - ()x-(1dv ;x u

dx (-2)x )x-(1x 232x-1xsen x arcx dxsen x rca x4

dx (-2)x )x-(1.x2132x-1xsen x arcx dxsen x rca x4

dx )x-(1x3-dx xarcsenx3-x-1x sen x arcxdxsen x arc x

xdx-1x3-dx xarcsenx3-x-1xsen x arcxdxsen x arc x

2x-1 sen x arcx )2(3x2x-1 sen x arcx xdxsen x arc 3

x

2343

232

121

1212

212

212343

212343

212232343

2232343

3

22

3

3

][

+==

==

++=

=

==

∫=∫=

++=

+=

+=

+=

++=

∫

∫∫∫∫

∫ ∫∫∫ ∫∫∫∫

∫

∫

+

+