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Question:

$x \log x$

Solution:

Let $I=\int x \log x d x$

Taking log x as first function and x as second function and integrating by parts, we obtain

$I=\log x \int x d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x d x\right\} d x$

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

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

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