求∫x^2lnxdx

∫x^2lnxdx
=1/3*∫lnxdx^3
=x^3lnx/3-1/3*∫x^3*(1/x)dx
=x^3lnx/3-1/3*∫x^2dx
=(x^3lnx)/3-x^3/9+C

求∫x^2lnxdx

解:∫x^2lnxdx
=1/3×∫lnxdx^3
=1/3×(x^3×lnx-∫x^3dlnx)
=1/3×(x^3×lnx-∫x^3×1/xdx)
=1/3×(x^3×lnx-∫x^2dx)
=1/3×(x^3×lnx-1/3∫x^2dx^3)
=x^3×lnx/3-x^3/9+c

应该是这样做...

令u=lnX,v'=x^2
则u'=1/X v=(1/3)X^3
由分部积分可得

∫x^2lnxdx
=lnX*(1/3)X^3-∫1/X 1/3*X^3dx
=lnX*(1/3)X^3-∫X^2/3dx
=lnX*(1/3)X^3-∫1dX^3
=lnX*(1/3)X^3-X^3+c

∫X2lnxdx=1/3 ∫lnxdx3=1/3x3lnx-1/3 ∫X3*1/xdx
=1/3x3lnx-1/3*1/3x3l
=1/3x3lnx-1/9x3