Originally Posted by

**harold** $\displaystyle

\int \frac {dx}{x ln x^2} dx $

1) $\displaystyle = \int \frac {dx}{2x ln x} dx $ This step follows from the property of logs (exponents)

2) $\displaystyle = \frac {1}{2}\int \frac {dx}{x ln x} dx $ Extract any constants and place in front of the integral

Let $\displaystyle u = ln x$, then $\displaystyle du = \frac {1}{x} dx. $

3) $\displaystyle = \frac {1}{2}\int \frac {1}{ln x} \cdot \bigg (\frac{1}{x}\bigg)dx $

4) $\displaystyle = \frac {1}{2}\int \frac {1}{u} du $ U-substitution applied

5) $\displaystyle = \frac {1}{2} ln u + C $ By Formula C, and $\displaystyle u > 0 $

6) $\displaystyle = \frac {1}{2} ln (ln x) + C$ By Reverse substitution