Question

Integrate the function w..r. to x : $\frac{\log(\log x)}{x}$

Answer 1

To solve this we must convert into integrable form,by substitution.
Let

$log\:x = t\\\\\frac{1}{x} dx= dt$

Now substitute the assumption

$\int\frac{log\:(log\:x)}{x} dx=\int log \:t \:dt$

Now Integration it by parts

$\int log \:t\: dt=log\: t\int 1\: dt-\int[\frac{d\: log t}{dt}]\:dt\\\\=t \:log\: t-t +C$

Now undo substitution

$\int\frac{log\:(log \:x)}{x} dx=log \:x(log\: log \:x)-log\: x +C$

or

$\int\frac{log\:(log \:x)}{x} dx=log \:x[(log\: log \:x)-1] +C$