Question

Integrate the function: $x\log 2x$

Answer 1

$\int \stackrel{\underset{}{x}}{II}\stackrel{\underset{}{log2x}}{I}dx$
Using by parts
$=\log 2x\int xdx-\int (\frac{d\left(\log 2x\right)}{dx}\int x.dx)dx$
$=\log 2x\left(\frac{x^2}{2}\right)-\int \frac{1}{2x}.2.\frac{x^2}{2}.dx$
$=\frac{x^2}{2}\log 2x-\int \frac{x}{2}.dx$
$=\frac{x^2}{2}\log 2x-\frac{1}{2}\int xdx$
$=\frac{x^2}{2}\log 2x-\frac{1}{2}\left(\frac{x^2}{2}\right)+C$
$=\frac{x^2}{2}\log 2x-\frac{x^2}{4}+C$
Where C is constant of integration