Question

Integrate the function $x(\log x{)}^2$

Answer 1

$I=\int x(\log x{)}^{2}dx$
Taking $(\log x{)}^2$ as first function and $1$ as second function and integrating by part, we obtain
$I=(\log x{)}^2\int xdx-\int [\left\{{\left(\frac{d}{dx}\log x\right)}^2\right\}\int xdx]dx$
$=\frac{x^2}{2}(\log x{)}^2-[\int 2\log x\cdot \frac{1}{x}\cdot \frac{x^2}{2}dx]$
$=\frac{x^2}{2}(\log x{)}^2-\int x\log xdx$
Again integrating by parts, we obtain
$I=\frac{x^2}{2}(\log x{)}^2-[\log x\int xdx-\int \{\left(\frac{d}{dx}\log x\right)\int xdx\}dx]$
$=\frac{x^2}{2}(\log x{)}^2-[\frac{x^2}{2}-\log x-\int \frac{1}{x}\cdot \frac{x^2}{x}dx]$
$=\frac{x^2}{2}(\log x{)}^2-\frac{x^2}{2}\log x+\frac{1}{2}\int xdx$
$=\frac{x^2}{2}(\log x{)}^2-\frac{x^2}{2}\log x+\frac{x^2}{4}+C$