Question

Integrate the function $x^2\log x$

Answer 1

Let $I=\int x^2\log xdx$
Taking $\log x$ as first function and $x^2$ as second function and integrating by parts, we obtain
$I=\log x\int x^{2}dx=\int \{\left(\frac{d}{dx}\log x\right)\int x^{2}dx\}dx$
$=\log x\left(\frac{x^3}{3}\right)-\int \frac{1}{x}\cdot \frac{x^3}{3}dx$
$=\frac{x^3\log x}{3}-\int \frac{x^2}{3}dx$
$=\frac{x^3\log x}{3}-\frac{x^3}{9}+C$