# How do you find the definite integral of int 1/(xlnx) from [e,e^2]?

Jan 24, 2017

The answer is $= \ln 2$

#### Explanation:

We do this integral by substitution

Let $u = \ln x$, $\implies$, $\mathrm{du} = \frac{\mathrm{dx}}{x}$

Therefore,

$\int \frac{\mathrm{dx}}{x \ln x} = \int \frac{\mathrm{du}}{u}$

$= \ln u = \ln \left(\ln x\right)$

So,

${\int}_{e}^{{e}^{2}} \frac{\mathrm{dx}}{x \ln x} = {\left[\ln \left(\ln x\right)\right]}_{e}^{{e}^{2}}$

$= \left(\ln \left(\ln {e}^{2}\right) - \ln \left(\ln e\right)\right)$

$= \ln 2 - \ln 1$

$= \ln 2$