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Find the integral of x log x

Evaluate the following integral:

    \[ \int x \log x \, dx. \]


We use integration by parts, defining

    \begin{align*}  u &= \log x & du &= \frac{1}{x} \, dx \\ dv &= x \, dx & v &= \frac{1}{2} x^2.  \end{align*}

Then we have,

    \begin{align*}  \int x \log x \, dx &= \int u \, dv \\  &= uv - \int v \, du \\  &= \frac{1}{2}x^2 \log |x| - \frac{1}{2} \int x \, dx \\  &= \frac{1}{2}x^2 \log |x| - \frac{1}{4} x^2 + C.  \end{align*}

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