Question

Integrate $\int x\log 2xdx$

Answer 1

$I=\int x\log 2xdx$

Let $u=\log 2x\Rightarrow du=\frac{1}{2x}\times 2dx=\frac{dx}{x}$

$dv=xdx\Rightarrow v=\frac{x^2}{2}$

$I=\frac{x^2}{2}\log 2x-\int \frac{x^2}{2}\frac{dx}{x}$

$=\frac{x^2}{2}\log 2x-\frac{1}{2}\int xdx$

$=\frac{x^2}{2}\log 2x-\frac{x^2}{4}+c$