Originally Posted by

**RedBarchetta** **Would anyone mind checking this to see if this looks correct? I'll explain more below:**

$\displaystyle

\int {\frac{{dx}}

{{x\log _{10} x}}}

$

** Simple enough, I make a substitution:**

$\displaystyle

\begin{gathered}

u = \log _{10} x \hfill \\

du = \frac{1}

{{\ln 10}} \cdot \frac{1}

{x}dx \hfill \\

dx = x\ln (10)du \hfill \\

\end{gathered}

$

$\displaystyle

\begin{gathered}

\int {\frac{{x\ln (10)du}}

{{x \cdot u}}} = \ln 10\int {\frac{1}

{u}du = \ln 10\left[ {\ln |u| + c} \right]} = \hfill \\

\ln 10\left[ {\ln |\log _{10} x| + c} \right] = \ln 10 \cdot \ln |\log _{10} x| + C \hfill \\

\end{gathered}

$

** I'm not sure how you would simplify the final result any further. **

This is what my book has for an answer. Am I still correct? Or did I go wrong somewhere?

$\displaystyle

\ln 10 \cdot \ln |\ln x| + C

$

** Thank you.**