Per partes integration: $\int x^2 \ln x dx$

Hint: Integrating by parts with $$u=lnx, dv=x^2dx$$

$$\int x^{2}\ln(x)\ dx=\left[\frac{1}{3}x^3\ln(x)\right]-\int \frac{1}{3}x^2dx=\frac{1}{3}x^3\ln(x) - \frac{1}{9}x^3 + C.$$