# What is int ln(lnx)/xdx?

Nov 3, 2015

Consider the following substitution.

Let:
$t = \ln x$
$\mathrm{dt} = \frac{1}{x} \mathrm{dx}$

$\int \ln \frac{\ln x}{x} \mathrm{dx} = \int \ln t \mathrm{dt}$

Now we can do an integration by parts.

$\int u \mathrm{dv} = u v - \int v \mathrm{du}$

Let:
$u = \ln t$
$\mathrm{du} = \frac{1}{t} \mathrm{dt}$
$\mathrm{dv} = \mathrm{dt}$
$v = t$

$= \ln t - \int t \cdot \frac{1}{t} \mathrm{dt}$

$= t \ln t - t$

And now let's substitute back in our original variables.

$= \textcolor{b l u e}{\left(\ln x\right) \ln \left(\ln x\right) - \ln x + C}$