La integración por partes

INTEGRACION POR PARTES

=xsenx+cosx+C

=xarc cos⁡〖2x- 1/2 √(1-4x^2 )〗+C

= -2/105 (1-x)^(3/2) (15x^2+12x+8)+C

= 1/2 (x^2+1) arctan⁡x- 1/2 x+C

= -2/3 cos^3 x-sen^2 x cos⁡x+C

= (2(bx-2a)√(a+bx))/(3b^2 )+C

= 1/2 x^2 arc sen x^2+ 1/2 √(1-x^4 )+C

= 1/2 x(sen ln⁡x-coslnx)+C

= (e^ax (a senbx-b cosbx))/(a^2+b^2 )+C

=x/(4(4+x^2 )^(1/2) )+C

= 1/2048 {(x(3x^2-80))/((x^2-16)^2 )+3/8 ln|(x-4)/(x+4)|}+C

= 3/8 x-3/8 senx cos⁡x-1/4 sen^3 xcosx+C

= -1/5 cos^3 x(sen^2 x+2/3)+C

INTEGRALES TRIGONOMETRICAS

∫▒〖cos^2 x dx= 1/2 x+ 1/4 sen 2x+C〗

= 1/6 cos^2 2x- 1/2 cos2x+C

∫▒〖〖(sen 2x)〗^4 dx=3/8 x-1/8 sen 4x〗+1/64 sen8x+C

= 3/8 x+1/2 senx+ 1/16 sen2x+C

∫▒〖〖(sen x)〗^7 dx=1/7 〖(cosx)〗^7-8/5 (cosx)^5+(cosx)^3-cosx+C〗

=5/16 x+1/2 senx+3/32 sen2x-1/24 sen^3 x+C

∫▒〖(senx)^2 (〖cosx)〗^5 dx=1/3 (senx)^3-2/5(〖senx)〗^5+1/7 (senx)^7+C〗

= 1/5 cos^5 x-1/3 cos^3 x+C

∫▒〖〖(senx)〗^3 〖(cosx)〗^3 dx=1/48 〖(cos2x)〗^3-1/16 cos2x+C〗

=1/128 (3x-sen4x+ 1/8 sen8x)+C

∫▒〖sen2x cos4x dx=1/4 cos2x-1/12 cos6x+C〗

= 1/2 senx+ 1/10 sen5x+C

∫▒〖sen 5x sen x dx=1/8 sen 4x-1/12 sen 6x+C〗

=senx+ 1/2 sen^2 x+C

∫▒〖〖(cosx)〗^(2⁄3)/〖(senx)〗^(8⁄3) dx〗 = -3/5 〖cot〗^(5⁄3) x+C

=csc⁡〖x- 1/3 csc^3 x+C〗

∫▒〖x(cos^3 x^2-sen^3 x^2 )dx=1/12 (senx^2+cosx^2 )(4+sen2x^2 )+C〗

= 1/2 tan^2 x+ln|cosx|+C

∫▒〖tag^3 3x sex3xdx=1/9 sec^3 3x-1/3 sec3x+C〗

= 2/5 tan^(5⁄2) x+ 2/9 tan^(9⁄2) x+C

∫▒〖tag^4 xsec^4 xdx=1/7 tag^7 x+1/5 tag^5 x+C〗

= -1/2 cot^2 x-ln|senx|+C

∫▒〖cot^3 〗 xcsc^3 xdx=-1/5 csc^5 x+1/3 csc^3 x+C

= -1/5 csc^5 x+ 1/3 csc^3 x+C

∫▒〖csc^4 2xdx=-1/2 cot2x-1/6 cot^3 2x+C〗

= - 1/(3tan^3 x)-1/tanx+C

∫▒(cot^3 x)/csc⁡x dx =-senx-cscx+C

=2√secx+C

INTEGRACION POR DESCOMPOCISION EN FRACCIONES SIMPLES

=1/5 ln|(x+1)/(x+6)|+C

=x+ln|(x+2)〖(x-4)〗^4 |+C

=ln⁡|x-2|- 2/(x-2)+C

=ln⁡|x/√(x^2+1)|+C

= 1/2 x^2+ln⁡〖|x/√(x^2-2x+3)|+C〗

=ln⁡〖(x^2+4)+1/2 arc tan⁡〖1/2 x+ 4/(x^2+4)〗+C〗

=ln |(x^3-x^2+x)/〖(x+1)〗^2 |- 3/(x+1)+ 2/√3 arc tan⁡〖(2x-1)/√3+C〗

= 1/(x^2+x+2)-3/(x^2+1)+ln⁡〖(x^2+1)/(x^2+x+2)〗+C

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