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Integrate the function
$x(\log x)^2$
$x(\log x)^2$
Solution:
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Verified Answer
$\int x(\log x)^2 d x$
$=\frac{x^2}{2}(\log x)^2-\left[(\log x) \cdot \frac{x^2}{2}-\int \frac{1}{x} \cdot \frac{x^2}{2} d x\right]$
$=\frac{x^2}{2}(\log x)^2-\frac{x^2}{2} \log x+\frac{1}{4} x^2+C$
$=\frac{x^2}{2}(\log x)^2-\left[(\log x) \cdot \frac{x^2}{2}-\int \frac{1}{x} \cdot \frac{x^2}{2} d x\right]$
$=\frac{x^2}{2}(\log x)^2-\frac{x^2}{2} \log x+\frac{1}{4} x^2+C$
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