“Año del deber ciudadano”

UNIVERSIDAD NACIONAL DE SAN MARTÍN - TARAPOTO

FACULTAD DE ECOLOGÍA

ESCUELA ACADÉMICA PROFESIONAL DE INGENIERIA AMBIENTAL

DEPARTAMENTO ACADÉMICO DE CIENCIAS AMBIENTALES

EJERCICIOS PROPUESTOS DE ANÁLISIS MATEMÁTICO II.

“INTEGRACION POR PARTES E IONTEGRACION POR SUSTITUCION TRIGONOMETRICA”

ASIGNATURA : MATEMATICA II

MOYOBAMBA-SAN MARTIN

2007

INTEGRACION POR PARTES

1. ∫(x2-2x+5) dx sea: u=x2-2x+5 dv=e-xdx

(x2-2x+5 )(-e-x)-∫(2x-2)(-e-x)dx Du= (2x-2) dx v=-e-x

-x2e-x+2xe-x-5e-x+∫(2x-2)(e-x)dx

-x2e-x+2xe-x-5e-x-2xe-x+2e-x+2∫e-xdx sea: u=2x-2 dv= e-xdx

-x2e-x+2xe-x-5e-x-2xe-x+2e-x-2e-x+C du=2dx v=-e-x

-x2e-x-5e-x+C.

2.∫(x3-3x)e6xdx = ∫[(t/6)3-3(t/6)]etdt/6 sea :t=6x x=t/6

=1/1296(∫t3etdt)-1/12(∫tetdt) dt/6=dx

=1/1296[t3et-∫3t2etdt]-1/12∫ tetdt

=1/1296[t3et-3t2et+∫6tetdt]-1/12∫ tetdt

=1/1296[t3et-3t2et+6tet-∫6et]-1/12∫ tetdt u=t3 du=3t2dt...1 dv=etdt v=et

=1/1296[t3et-3t2et+6tet-6et]-tet/+et/12+C u=3t2 du=6tdt...2 u=t du=dt

=e6x/216[36x3-18x2-102x+17]+C. u=6t du=6dt...3

3.∫(3X2+2X-1)dx/4e3x =1/4∫(3X2+2X-1)(e-3x) z=-3x x=-z/3 dx=-dz/3

=-1/12{∫ [z2/3-2z/3-1] ezdz}

=-1/36{∫ [z2-2z-3] ezdz} u= z2-2z-3 du=(2z-2)dz dv= etdt

=-1/36[(z2-2z-3) (ez)-∫(2z-2) ezdz] u=2z-2 du=2dz

=-1/36{z2ez-2zez-3ez-[(2z-2) ez-∫2ezdz]}

=-1/36[z2ez-2zez-3ez-2zez+2ez+2ez+C]

=-1/36(9x2e-3x-12xe-3x+e-3x)+k

=-e-3x/12(3x2-4x+1/3)+k

4.∫(8x3+6x2+2x+5)e4x=1/4∫(z3/8+3z2/8+z/2+5)ezdz sea: z=4x dz/4=dx

=1/32∫z3ezdz+3/32∫z2ezdz+∫zezdz/8+5/4∫ezdz u=z3 du=3z2dz dv=ezdz v=ez

=1/32(z3ez-3∫ z2ezdz)+ 3/32∫z2ezdz+1/8∫zezdz+5/4∫ezdz

= z3ez/32+1/8∫zezdz+5ez/4+C

= z3ez/32+1/8(zez-∫ezdz) +5ez/4+C u=z du=dz dv=ezdz v=ez

= z3ez/32+zez/8-ez/8 +5ez/4+C

= (4x)3 e4x/32+4xe4x/8+9e4x/8)+C

=2x3e4x+xe4x/2+9e4x/8+C

= e4x (2x3+x/2+9/8)+C

5.∫arctag(x) 1/2dx

=2∫arctgtdt (x)1/2=t 2tdt=dx

=2(t2arctgt/2-1/2∫t2dt/t2+1) u=arctgt du=dx/1+t2

= t2arctgt-∫ (t2+1-1)dt/ t2+1 dv=tdt v=t2/2

= t2arctgt-t+arctgt+C

=xarctg(x)1/2-(x)1/2-arctg(x)1/2+C

6. ∫xarctg2xdx =x2arctg2x/2-∫x2arctgxcdx/x2+1 u= arctg2x du=2arctgxdx/ x2+1 dv=xdx v= x2/2

= x2arctg2x/2-∫x2+1arctgxdx/x2+1

= x2arctg2x/2-∫arctgdx+∫arctgxdx/x2+1 u=arctgx du=dx/ x2+1 dv=dx v=x

= x2arctg2x/2-(xarctgx-∫xdx/ x2+1)+∫arctgxdx/x2+1 t=arctgx dt= dx/ x2+1

= x2arctg2x/2-xarctgx+2ln|x2+1|+∫tdt

= x2arctg2x/2-xarctgx+2ln|x2+1|+arctg2x/2+C

=1/2[(x2+1) arctg2x-2 xarctgx+2ln|x2+1|+C

7. ∫arctgxdx/x2(1+x2) =∫zdz/tg2z z=arctgx x=tgz

=z(-ctgz-z)-∫(-ctg-z)dz dz=dx/1+x2 u=z du=dz dv=ctg2zdz

=-zctgz-z2+ln|senx|+z2/2 v=-ctgz-z

=-arcctg/x-arctg2x/2+ln|x/(x2+1)1/2|+C

8. ∫x2arctg3xdx = 1/27∫z2arctzdz z=3x x=z/3 dx=dz/3

=1/27(z3arctgz/3-1/3∫z3dz/z2+1) u=arctgz du=dz/z2+1

=1/81[z3arctgz-∫(z2+1-1)zdz/z2+1] dv=z2dz v=z3/3

=1/81[z3arctgz-∫zdz+∫zdz/ z2+1]

=1/81[z3arctgz-z2/2+ln|z2+1|+C]

=(3x)3arctg3x/81-(3x)2/162+ln|9x2+1|/162+k

=x3arctg3x/3-x2/18+ln|9x2+1|/162+k

9.∫(x2+1)ln(x+1/x-1)dx/(1-x2) =(1-x2)1/2ln(x+1/x-1)+∫2(1-x2)1/2dx/1-x2 u=ln(x+1/x-1) du=2dx/1-x2

=(1-x2)1/2ln(x+1/x-1)+2∫dx/(1-x2)1/2 dv=xdx/(1-x2)1/2 v =-(1-x2)1/2 /2

=(1-x2)1/2ln(x+1/x-1)+2arcsenx+C

10.∫arctg(x+1)1/2dx =2∫zarctgzdz z=(x+1)1/2 2zdz=dx

=2[z2arctgz/2-∫z2dz/2(z2+1) ]

= z2arctgz-∫(z2+1-1)dz/z2+1 u=arctgz du=dz/z2+1

= z2arctgz-∫dz+∫dz/ z2+1 dv=zdz v=z2/2

= z2arctgz-z+arctgz+C

=(x+1)arctg(x+1)1/2- (x+1)1/2+arctg(x+1)1/2+C

= [x+2]arctg(x+1)1/2- (x+1)1/2+C

11. ∫xarctg(x2-1)1/2dx =x2arctg(x2-1)1/2/2-∫x2dx/2x(x2-1)1/2= u=arctg(x2-1)1/2 du=dx/x(x2-1)1/2

= x2arctg(x2-1)1/2/2-1/2∫xdx/(x2-1)1/2 u dv=x v=x2/2

= x2arctg(x2-1)1/2/2-1/4∫dt/t1/2

= x2arctg(x2-1)1/2/2 -(x2-1)1/2/2+ C

12. ∫xarctagxdx/(1+x2)2 =-arctgx/2(1+x2) +∫dx/2(1+x2)2 u=arctgx du=dx/x2+1 dv=xdx/(x2+1)2 v=-1/2(1+x2)

=-arctgx/2(1+x2) +∫dx/2[(1+x2)1/2]4 x=tgӨ dx=sec2Өdx

= -arctgx/2(1+x2) +1/2(∫sec2ӨdӨ/sec4Ө) (1+x2)1/2=secӨ

=-arctgx/2(1+x2) +1/2∫cos2ӨdӨ

=-arctgx/2(1+x2) +1/4∫(1+cos2Ө)dӨ

=-arctgx/2(1+x2) +arctgx/4+2senӨcosӨ/8+k

=-arctgx/2(1+x2) +arctgx/4+x/4(1+x2)+k

13. ∫arctgx1/2dx/x1/2 =∫arctgt(2tdt)/t t=x1/2 t2=x 2tdt=dx

=2∫arctgtdt u=arctgt du=dt/t2+1 dv=dt v=t

=2[tarctgt-∫tdt/t2+1] z= t2+1 dz/2=tdt

=2tarctgt-2∫dz/2z

=2tarctgt-ln|z|+k

=2x1/2arctgx1/2-ln|x+1|+k

14. ∫xsec2xdx =xtgx-∫tgxdx u=x du=dx dv=sec2xdx v=tgx

=xtgx+ln|cosx|+C

15.∫xtg2xdx =x(tgx-x)-∫(tgx-x)dx u=x du=dx dv=tg2xdx v=tgx-x

=xtgx-x2-∫tgxdx+∫xdx

= xtgx-x2+ln|cosx|+x2/2+k

=xtgx-x2/2+ln|cosx|+k

16. ∫senx1/3dx =3∫z2senzdz z=x1/3 x3=z 3z2dz=dx u=z2 du=2zdz

=3[-z2cosz-∫(-cosz)(2zdz)] dv=senzdz v=-cosz

=-3z2cosz+6∫zcoszdz u=z du=dz dv=coszdz v=senz

=-3z2cosz+6(zsenz-∫senzdz)

=-3z2cosz+6zsenz+6cosz+k

=-3x2/3cosx1/3+6x1/3senx1/3+6cosx1/3+k

=3[cosx1/3(2-x2/3)+2x1/3senx1/3]+k

17. ∫x3senxdx =-x3cosx-∫(-cosx)(3x2dx) u=x3 du=3x2dx dv=senxdx v=-cosx

=-x3cosx+∫x2cosxdx u=x2 du=2xdx dv=cosxdx v=senx

=-x3cosx+3(x2senx-2∫xsenxdx)

=-x3cosx+3x2senx-6∫xsenxdx u=x du=dx dv=senxdx v=-cosx

=-x3cosx+3x2senx-6(-xcosx-∫-cosxdx)

=-x3cosx+3x2senx+6xcosx-6senx+k

18. ∫(x2+5x+6)cos2xdx =∫[(z/2)2+5z/2+6)]cosz(dz/2) z=2x x=z/2 dx=dz/2

=1/8∫z2coszdz+5/4∫zcoszdz+3∫cozdz u=z2 du=2zdz dv=coszdz v=senz

=1/8(z2senz-2∫zsenzdz)+5/4∫zcoszdz+3∫cozdz u=z du=dz dv=senzdz v=-cosz

=z2senz/8-1/4∫zsenzdz+5/4∫zcoszdz+3senz u=z du=dz dv=coszdz v=senz

=z2senz/8-1/4(-zcoz-∫-cozdz)+5/4(zsenz-∫senzdz)+3senz

=z2senz/8+zcosz/4-senz/4+5zsenz/4+5cosz/4+3senz+k

=z2senz/8-senz/4+5zsenz/4+3senz+zcosz/4+5cosz/4+k

=x2sen2x/2-sen2x/4+10xsen2x/4+3sen2x+2xcos2x/4+5cos2x/4+k

=sen2x(2x2+10x+11)/4+cos2x(2x+5)/4+k

19. ∫xsec23xdx =1/9∫zsec2zdz z=3x x=z/3

=1/9(ztagz-∫tgzdz) u=z du=dz dv=sec2zdz v=tgz

=ztgz/9-ln|senz|/9+k

=3xtg3x/9-ln|sen3x|/9+k

=xtg3x/9-ln|sen3x|/9+k

20. ∫xcsc2(x/2)dx =4∫zcsc2zdz x=z/2 x=2z dx=2dz

=4(-zctgz-∫-ctgzdz) u=z du=dz dv=csc2zdz v=-ctgz

=-4zctgz+4ln|senz|+k

=-4(x/2)ctg(x/2)+4ln|sen(x/2)|+k

=-2xctg(x/2)+4ln|sen(x/2)|+k

21. ∫xdx/sen2x =∫xcsc2xdx u=x du=dx dv=csc2xdx v=-ctgx

=-xctgx+∫ctgxdx

=-xctgx+ln|senx|+C

22. ∫9xtg23xdx =9∫ttg2tdt/9 t=3x x=t/3 dt/3=dx

=∫ttg2tdt u=t du=dt dv=tg2tdt v=tgt-t

=t(tgt-t)-∫(tgt-t)dt

=ttgt-t2-∫tgtdt+∫tdt

=ttgt-t2+ln|cot|+t2/2+k

=3xtg3x-9x2+ln|cos3x|+9x2/2+k

=3xtg3x-9x2/2+ln|cos3x|+k

23. ∫x2senxdx =-x2cosx+2∫xcosxdx u=x2 du=2xdx dv=senxdx v=-cosx

=-x2cosx+(2xsenx-2∫senxdx) u=2x du=2dx dv=cosxdx v=senx

=-x2cosx+2xsenx+2cosx+k

24. ∫xcos3xdx =xsen3x/3-∫sen3xdx/3 u=x du=dx dv=cos3xdx v=sen3x/3

=xsen3x/3+cos3x/9+C

25. ∫xcosxdx/sen2x =-x/senx+∫dx/senx u=x du=dx dv=cosxdx/sen2x

=-x/senx+∫cscxdx v=-1/senx

=-x/senx+ln|cscx-ctgx|+C

26. ∫sen(2x)1/2dx =∫tsentdt t=(2x)1/2 tdt=dx

=-tcost+∫cotdt u=t du=dt dv=sentdt

=-tcost+sent+C v=-cost

-(2x)1/2cos(2x)1/2+sen(2x)1/2+C

27. ∫xnlnxdx ;n≠-1 =xn+1lnx/n+1-∫xn+1dx/x(n+1) u=lnx du=dx/x dv=xn

= xn+1lnx/n+1-1/n+1∫xndx v=xn+1/n+1

= xn+1lnx/n+1-xn+1/(n+1)2

28. ∫ln3xdx/x2 =-ln3x/x+3∫ln2xdx/x2 u=ln3x du=3ln2xdx/x dv=dx/x2 v=-1/x

=-ln3x/x+3[-ln2x/x+2∫lnxdx/x2] u=ln2x du=2lnxdx/x dv=dx/x2 v=-1/x

=-ln3x/x-3ln2x/x+6(-lnx/x+∫dx/x2) u=lnx du=dx/x dv=dx/x2 v=-1/x

=-ln3x/x-3ln2x/x-6lnx/x-6/x+k

=-1/x(ln3x+3ln2x+6lnx+6)+k

29. ∫ln2xdx =xln2x-2∫lnxdx u=ln2x du=2lnxdx/x

=xln2x-2(xlnx-x)+C dv=dx v=x

=xln2x-2xlnx-2x+C

=x(ln2x-2lnx+2)+C

30. ∫ln(cosx)dx/cos2x =tgxln(cosx)+∫ctg2xdx u=ln(cosx) du=-tgxdx

=tgxln(cosx)+tgx-x+c dv=dx/cos2x v=tgx

31. ∫(x2-2x+3)lnxdx =∫x2lnxdx-2∫xlnxdx+3∫lnxdx u=lnx du=dx/x dv=x2dx v=x3/3

=x3lnx/3-1/3∫x2dx-2∫xlnxdx+3∫lnxdx u=lnx du=dx/x dv=xdx v=x2/2

=x3lnx/3-x3/9-2[x2lnx/2-1/2∫xdx]+3(xlnx-x)+C

= x3lnx/3-x3/9-x2lnx+x2/2+3xlnx-3x+C

=lnx(x3/3-x2+3x)-x3/9+x2/2-3x+C