Evaluate the integral: $\int{\frac{1}{x\sqrt{\ln{x}}}}.dx$

Question

I am looking to evaluate the following integral:

$$\int{\frac{1}{x\sqrt{\ln{x}}}.dx}$$

but I cannot figure out how to solve it by substitution or by parts. Using the integration by parts, I separated the equation as follows:

$$\int{\frac{1}{x}.\frac{1}{\sqrt{\ln{x}}}.dx}$$

but I keep getting even more complicated equations. Any suggestions on solving this?

Answer 1

$$\int { \frac { 1 }{ x\sqrt { \ln { x } } } dx } =\int { \frac { d\left( \ln { x } \right) }{ \sqrt { \ln { x } } } = } 2\sqrt { \ln { x } } +C$$

Answer 2

$$\int\frac{dx}{x\sqrt{\ln x}}$$ Take $x=e^u$. Then $$=\int \frac{e^{-u}}{\sqrt u}\cdot e^u du$$ $$=\int \frac{du}{\sqrt u}$$ $$=2\sqrt{u}+C$$ $$=2\sqrt{\ln(x)}+C$$ And there you go!