Integrales

INTEGRALES INMEDIATAS
FORMULAS:
1.- dx=x+c
2.- du+dv-dw=du+du-dw
3.- adv=adv
4.- vndv=vn+1n+1+c
5.- dvv=ln v+c
6.- avdv=avlna+c
7.- evdv=ev+c
8.- sen v dv=-cos v+c
9.- cosv dv=sen v+c
10.- tanv dv=lnsecv+c
11.- cotv dv=lnsen v+c
12.- secv dv=lnsec v+tanv+c
13.- cscv dv=lncscv-cotv+c
14.- sec2v dv=tanv+c
15.- csc2v dv=-cotv+c
16.- secv ∙tanvdv=secv+c
17.- cscv∙cotvdv=-cscv+c

IDENTIDADES TRIGONOMÉTRICAS

tan2x=sec2x-1 cot2x=csc2x-1 | sen2x=1-cos2x cos2x=1-sen2x sec2x=tan2x+1 csc2x=cot2x+1 |
sen2x=1-cos2x2 | cos2x=1+cos2x2 |

sen x∙cosx=12∙sen 2x
tanx=sen xcosx cotx=cosxsen x
cotx=1tan x , secx=1cos x , cscx=1sen x

EJERCICIOS
1. 5dx
=5dx
=5x+c

2. a dx
=adx
=ax+c

3. 15dx
=15dx

=15x+c

4. a+bdx=a+bdx
=a+bx+c

5. x dx v=x n=1 dv=dx
=x22+c =x22=c
6. x2dx v=x n=2 dv=dx
=x33+c
7. x5dx =x66 9. 1x2dx=x-2dx =x-1-1+c 11. 1xndx =xn dx =x-n+1-n+1+c 13. 3x dx=x1/3dx =x4/343+c =3x4/34+c 15. 1xdx=1x1/2dx =x-1/2dx =x1/212+c =2x+c | 8. xndx=xn+1n+1+c 10. 1x3dx =X-3dx =x-2-2+c =-12x2+c 12. x dx=x1/2dx =x3/232+c=2x3/23+c 14. nxdx=x1+nn1+nn+c =nx1+nn1+n+c 16. 13x dx =x-1/3dx =x2/323+c =33x22+c |

17. 1nxdx=x-1n dx =xn-1nn-1n+c =n xn-1nn-1+c 19. ax2+bx+c dx =ax2dx+bx dx+c dx =ax2dx+bxdx+cdx =ax33+bx22+cx+c 20. 5x2-7x3dx=5x2 dx-7x-3dx =5x2 dx-7x-3dx =5x33-7x-2-2+c =5x33-72x2+c 22. 5x5-3xdx=5x5-3x-1/2dx =5x66-3x1212+c =5x66-3x+cn | 18. 2x3-5x2dx=2x3dx-5x2dx =2x3dx-5x2dx =2x44-5x33+c =x42-5x33+c 19’. 3x-42dx=9x2-24x+16dx =9x2dx-24xdx+16dx =9x33-24x22+16x+c =3x3-12x2+16x+c 21. a+b3x2dx =a+bx2/3dx =a dx+bx2/3dx =adx+bx2/3dx =ax+3bx5/3+c 23. 6x3+5x2-4x2dx=6x dx+5dx-4x-2dx =6x dx+5dx-4x-2dx =6x22+5x-4x-1-1+c =3x2+5x+4x+c |

24. 4x2-2xxdx=4x-2xxdx =4x-2x-1/2dx =4x dx-2x-1/2dx =4x22-2x1/212+c =2x2-4x+c 26.x2+12dx=x4+2x2+12dx =x4dx+2x2dx+dx =x55+2x33+x+c | 25. x3x-2dx =x1/23x-2dx =3x3/2dx-2x1/2dx =3x3/2dx-2x1/2dx =3x5/252-2x3/232+c =6x5/25-4x3/23+c |

CAMBIO DE VARIABLE
De aquí en adelante no nos ocuparemos únicamente de funciones de una sola variable, para funciones más generales hay que ver si la integral está completa, en caso contrario hay que completarla por el cambio de variable.
1.4+2x2dx v=4+2x n=2
=124+2x22dx dv=2 dx
=124+2x36+c Falta en la integral un 2.
=4+2x36+c

2. 3x2+42x dx v=3x2+4 n=2
=163x2+426x dx dv=6x dx
=16 3x2+433+c Falta en la integral un 6.
=3x2+4318+c
3. bx3dx v=bx n=3
=1bbx3b dx dv=b dx
=1bbx44+c
=bx44b+c

4. 6x dx v=6x n=1/2
=6x1/2 dx dv=6 dx
=166x1/2 6 dx
=166x3/232+c
=26x318+c

5.a+bx dx
=a+bx1/2dx v=a+bx n=1/2
=1ba+bx1/2b dx dv=b dx
=1b a+bx3/232+c
=2a+bx33b+c
6. dxa-bx
=a-bx-1/2dx
=-1b a-bx1/212-b dx
=-1ba-bx1/212+c
=-2a-bxb+c
7. dx4-x4 v=4-x n=-4
=4-x-4dx dv=-dx
=4-x-4-dx Falta en la integral un -1.
=-4-x-3-3+c
=14-x3+c
8. dxa+bx1/4
=a+bx-1/4dx v=a+bx n=-1/4
=1ba+bx-1/4b dx dv=b dx
=1ba+bx3/434+c=4a+bx3/43b+c
10. x3+a33x2dx
=3x3+a3x2dx v=x3+a n=3
=33x3+a33x2dx dv=3x2dx
=x3+a44+c
10’. 3x42x5+93dx
=3402x5+94+c
11. 2+x22x dx v=2+x2 n=2
=122+x222x dx dv=2x dx
=122+2x233+c
=2+x236+c

12. 5-bx22x dx v=5-bx2
=-12b5-bx24-2b dx dv=-2bx dx
=-12b5-bx255+c Falta la integral -2b
=-5-bx2510b+c
13. x3+35x2dx v=x3+3
=13x3+353x2dx dv=3x2 dx
=x3+3618+c14. x dxx2+84
=x2+8-4x dx
=12x2+8-42x dx
=-x2+8-3-6+c
=16x2+83+c
15. 1+x4 x3dx
=1+x41/2x3dx
=141+x41/24x3dx
=141+x43/232+c
=1+x1/43/2x3/4dx
16. 1+x1/45x3/4dx v=1+x1/4 n=5
=1+x1/45x-3/4dx dv=14x-3/4dx
=1141+x1/4514x-3/4dx Falta en la integral 14
=41+x1/466+c
=21+x1/463+c
17.1+x2xdx v=1+x1/2 n=2
=1+x1/22x1/2dx dv=12x-1dx
=1+x1/22x-1/2dx...