Um dieses Integral zu berchnen benutzen wir die partielle Integration.
$$\int x^2\ln (x) dx=\int \left(\frac{x^3}{3}\right) '\ln (x)dx \\ =\frac{1}{3}\int \left(x^3\right) '\ln (x)dx \\ =\frac{1}{3}\left(x^3\ln (x)-\int x^3\left(\ln (x)\right)'dx\right) \\ =\frac{1}{3}\left(x^3\ln (x)-\int x^3\cdot \frac{1}{x}dx\right) \\ =\frac{1}{3}\left(x^3\ln (x)-\int x^2dx\right) \\ =\frac{1}{3}\left(x^3\ln (x)-\frac{x^3}{3}+c\right)\\ =\frac{x^3\ln (x)}{3}-\frac{x^3}{9}+C$$ wobei C=c/3.
