Evaluate the following integrals:

Let $I=\int \frac{\log (\log x)}{x} d x$

It can be written as, $=\int\left(\frac{1}{x}\right)(\log (\log x)) d x$

Using integration by parts,

$I=\log (\log x) \int \frac{1}{x} d x-\int\left(\frac{1}{x \log x} \int \frac{1}{x} d x\right) d x$

We know that, $\int \log x=\frac{1}{x}$ and $\frac{d}{d x} \frac{1}{x}=\log x$

$=\log x(\log x) \times \log x-\int \frac{1}{x \log x} \times \log x d x$

$=\log x(\log x) \times \log x-\int \frac{1}{x} d x$

$=\log x(\log x) \times \log x-\log x+c$

$=\log x(\log (\log x)-1)+c$