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Find the integral of x2 log2x

by RoRi

Evaluate the following integral:

    \[ \int x^2 \log^2 x \, dx. \]

First, we want to integrate by parts, so define

    \begin{align*} u &= \log^2 x & du &= \frac{2 \log x}{x} \\ dv &= x^2 \, dx & v &= \frac{1}{3} x^3. \end{align*}

Then, we integrate by parts to obtain

    \begin{align*} \int x^2 \log^2 x \, dx &= \int u \, dv \\ &= uv - \int v \, du\\ &= \frac{1}{3}x^3 \log^2 |x| - \frac{2}{3} \int x^2 \log x \, dx \end{align*}

From the previous exercise (Section 6.9 Exercise #22) we know that for n \neq -1

    \[ \int x^n \log (ax) \, dx = \frac{x^{n+1}}{n+1}\log |ax| - \frac{x^{n+1}}{(n+1)^2} + C. \]

Since we want to evaluate the integral of x^2 \log x we have n = 2 and a = 1. Therefore, by the formula,

    \[ \int x^2 \log x \, dx = \frac{x^3}{3} \log |x| - \frac{x^3}{9} + C. \]

Now, plugging this back into the formula above,

    \begin{align*} \int x^2 \log^2 x \, dx &= \frac{1}{3}x^3 \log^2 |x| - \frac{2}{3} \int x^2 \log x \, dx \\ &= \frac{1}{3} x^3 \log^2 |x| - \frac{2}{3} \left( \frac{x^3}{3} \log |x| - \frac{x^3}{9} \right) + C \\ &= \frac{1}{3} x^3 \log^2 |x| - \frac{2x^3}{9} \log |x| - \frac{2x^3}{27} + C \\ &= \frac{x^3}{3} \left( \log^2 |x| - \frac{2}{3} \log |x| + \frac{2}{9} \right) + C. \end{align*}

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