Improper Integral $\int_{1/e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $

Question

I need some advice on how to evaluate it. $$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $$ Thanks!

Answer 1

Here's a hint: $$ \int_{1/e}^1 \frac{1}{\sqrt{\ln x}} {\huge(}\frac{dx}{x}{\huge)}. $$ What that is hinting at is what you need to learn in order to understand substitutions. It's all about the chain rule. The part in the gigantic parentheses becomes $du$.

Answer 2

To recap all that happened in the comments section:

Based on the initial problem of$$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}}$$

We perform a u-substitution with $u=\ln{x}$ and $du=\frac{dx}{x}$. Also, the bounds are converted to $\ln\frac{1}{e}=-1$ and $\ln{1}=0$. So we have:

$$\int\limits_{-1}^{0} \frac{du}{\sqrt{u}}=2\sqrt{u}\big|_{-1}^{0}=0-2i=-2i$$

And that's the solution!