Question

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

Answer 1

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

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