Answer
$\ln 2\approx 0.693$
check with desmos online calculator:
Work Step by Step
Find the indefinite integral first,
$\displaystyle \int\frac{\mathrm{l}}{x\ln x}dx=\int(\frac{1}{\ln x})\frac{1}{x}dx$
$\left[\begin{array}{l}
u=\ln x\\
du=\frac{1}{x}dx
\end{array}\right]$
$=\ln|u|+C$
$=\ln|\ln x|+C$
Now, the definite integral:
$\displaystyle \int_{e}^{e^{2}}\frac{\mathrm{l}}{x\ln x}dx=[\ln|\ln|x||]_{e}^{e^{2}}$
$=\ln|\ln e^{2}|-\ln|\ln e|$
$=\ln 2-\ln 1$
$=\ln 2\approx 0.693$
