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