Evaluating $\int \frac{\operatorname dx}{x\log x}$

Question

How to integrate $\frac{1}{x\log x}$?

Could you give me some ideas on how to integrate this? thanks.

i've tried setting $u=(\log x)^{-1}$.

$\dfrac{\mathrm du}{\mathrm dx} = x^{-1}$

But it didnt work...

Answer 1

Hint: For all $x>1$, $\dfrac 1{x\log (x)}=\dfrac{1/x}{\log (x)}$.

Answer 2

Hint. Use the substitution $u=\log(x)$.

Answer 3

Let : $$u = \log{x}$$ Then, $$\frac{du}{dx} = \frac{1}{x}$$ $$du = \frac{1}{x}dx$$ Hence, $$\begin{align}\int\frac{1}{x\log{x}}dx &=\int\frac{1}{\log{x}}\frac{1}{x}dx\\&= \int\frac{1}{u}du \\&= \log{|u|} + C \\&=\log|\log{u}| + C\end{align}$$