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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?

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  • $\begingroup$ Hint: Use the substitution $$\ln(x)=t$$ $\endgroup$
    – Nishant Garg
    Commented Aug 26, 2017 at 14:06
  • $\begingroup$ No \large, please. $\endgroup$
    – Did
    Commented Aug 26, 2017 at 14:14

2 Answers 2

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$$\int\frac{dx}{x\sqrt{\ln x}}$$ 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!

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

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