Find the indefinite integral using integration by parts with the given choices of u and dv. (Use C for the constant of integration.) x2 ln(x) dx; u

∫ x²Lnx dx   =  1/3 [ x³Lnx - (1/3)x³] + C

Step-by-step explanation:

∫ x²Lnx dx

Integration by parts:

if we have    u*v    then   D(u*v)  =  v*du  +  u*dv   (1)

We make changes of variables :

Lnx dx  = du           then    u  =  xLnx - x

v  = x²                      then   dv =  2xdx

And    

∫ x²Lnx dx    becomes      ∫vdu

According to expression (1)

∫vdu  =  u*v  -  ∫udv

Now by substitution

∫vdu  =  x² ( xLnx - x )  - ∫( xLnx - x) 2xdx

∫ x²Lnx dx  =  x²  ( xLnx - x )  - ∫ 2x²Lnxdx + ∫2x²dx

∫ x²Lnx dx  = x²  ( xLnx - x ) - 2 ∫x²Lnxdx  + 2 (x³/3) + C

∫ x²Lnx dx  + 2 ∫x²Lnxdx  = x²  ( xLnx - x ) + 2 (x³/3) + C

3 ∫ x²Lnx dx   = x³Lnx -x³ + 2/3)x³  +C

∫ x²Lnx dx   =  1/3 [ x³Lnx - (1/3)x³] + C