# Evaluate the integral. \int x^{2} \ln x dx

Evaluate the integral.
$\int {x}^{2}\mathrm{ln}xdx$
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Corben Pittman
Apply integration by parts:
$\int udv=uv-\int vdu$
Let, $u=\mathrm{ln}x$
$v=\frac{{x}^{3}}{3}$
${v}^{\prime }={x}^{2}$
$dv={x}^{2}dx$
$\int {x}^{2}\mathrm{ln}xdx=\frac{1}{3}{x}^{3}\mathrm{ln}\left(x\right)-\int \frac{{x}^{2}}{3}dx$
$=\frac{1}{3}{x}^{3}\mathrm{ln}\left(x\right)-\frac{{x}^{3}}{9}+C$