Evaluate x x d x - Vidyadip

Question

Evaluate $\int x \log x d x$

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Answer

$\text { Let } I =\int x \cdot \log x d x$
$=\log x \int x d x-\int\left[\frac{ d }{ d x}(\log x) \int x d x\right] d x$
$=\log x \cdot \frac{x^2}{2}-\int\left[\frac{1}{x} \times \frac{x^2}{2}\right] d x$
$=\frac{x^2}{2} \log x-\frac{1}{2} \int x d x$
$=\frac{x^2}{2} \log x-\frac{1}{2} \cdot \frac{x^2}{2}+ c$
$\therefore I =\frac{x^2}{2} \log x-\frac{x^2}{4}+ c $

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