Ejercicios derivadas e integrales

fx=4x2-3x2+2
y´ = f´x = 4x2´-3x2'+2´ = 12x2-6x

fx=3x22x+3
y'=f'x=3x2'2x+3-3x2(2x+3)'(2x+3)2=6x2x+3-3x2(2)(2x+3)2=12x2+18x-6x24x2+12x+9
=6x2+18x4x2+12x+9

fx=3x∙senxy'=f'x=3x'senx+3xsenx'=x13'senx+3xsenx'
y'=13x13-33senx+3xcosx=x-233senx+3xcosx=senx33x2+3x cosx

fx=x lnx
y'=f'x=x'lnx+xlnx' = 1lnx+x1x = lnx+ xx = lnx+1

fx=senx+12x-3y'=f'x=cosx+12x-3∙x+12x-3'=cosx+12x-3∙x+1'2x-3-x+1(2x-3)'(2x-3)2
y'=cosx+12x-3∙12x-3-x+122x-32=cosx+12x-3∙2x-3-2x-22x-32
y'=cosx+12x-3∙-52x-32=cosx+12x-3∙ -54x2-12x+9
3x-1 dx
umdu ;u=3x-1 ; dudx=3 ∴ du=3 dx ; m=12
133x-1 3dx=13(3x-1)12 3=133x-112+2212+22=133x-13232=2(3x-1)329+C

2-3x dx
umdu ;u=2-3x ; dudx=3 ∴ du=-3 dx ; m=12
-132-3x -3dx=-132-3x12-3dx=-132-3x12+2212+22=-132-3x3232
=-2(2-3x)329+C

(2x2+3)13 x dxumdu ; u=2x2+3 ; dudx=4 ∴ du=4 dx ; m=13
14(2x2+3)13 4x dx=14∙(2x2+3)13+3313+33=14∙(2x2+3)4343=3(2x2+3)4316+Ce-x2+2 x dx
u=x ; du=dx ; v=e-x2+2 ; dv=e-x2+2 ; u dv=u v- v du
= e-x2+2 x-e-x2+2 dx = e-x2+2 x- e-x2+2+C = e-x2+2 x-1+Cx2ex3dx
u=x2 ; du=2x dx ; v=ex3 ; dv= ex3 ; u dv=u v- v du
=x2ex3 -exe 2x dx
u=2x ; du=2 dx ; v=ex3; dv= ex3 ; u dv=u v- v du
= x2ex3- 2x ex3 –exe 2x dx = x2ex3- 2x ex3- 2 ex3 dx
= x2 ex3- 2x ex3+ 2 ex3+ C = ex3x2-2x+2+C

(ex+1)2dx
umdu ;u=ex+1 ; dudx=ex ∴ du=ex dx ; m=1ex

1exex+12 ex dx = 1ex ∙ (ex+1)2+12+1 = 1ex ∙ (ex+1)33 = (ex+1)33ex+ C