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Find the integral of (log x)2

by RoRi

Evaluate the following integral:

    \[ \int (\log x)^2 \, dx. \]

We use integration by parts, defining

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

First, we recall that \int \log x \, dx was computed in Example 2 on p. 235 of Apostol which gave

    \[ \int \log x \, dx = x \log |x| - x + C. \]

Then we have,

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

7 comments

  2. Artem says:

    Why does the answer put an absolute sign in the answer? I think there is a mistake in Apostol answers – the answer should be clearly in terms of log(x) instead of log|x|, since x>0 due to the way the problem is defined. \log^2(x) cannot accept negative x by default, only \log(|x|) can. Thus, the integration bounds have to be strictly positive. Also the derivation of Apostol on page 235 is in terms of x, not |x|.

  4. Nitesh Pandey says:

    Not satisfied by this solution please solve it in easy way that can be understood easily using formula
    ( “|” it willl be the sign of integration because there is no actual sign of integratio in mobile phones so please be understood)
    Formula:-
    u|vdx-(|du/dx |vdx) dx
    This formula is given in elements maths of 12th class.
    Thank You
    Nitesh Pandey

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