Question

Integrate the function $x\log 2x$

Answer 1

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