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Find the integral of (log |x|) / (x (1+log |x|)1/2)

by RoRi

Evaluate the following integral:

    \[ \int \frac{\log |x|}{x \sqrt{1+\log |x|}} \, dx. \]

To evaluate the integral, first we make the substitution,

    \[ u = 1 + \log |x|, \qquad du = \frac{1}{x} \, dx. \]

Therefore we have,

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

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