Answer
$\int_{e}^{e^4}\frac{dx}{x\sqrt{ln~x}} = 2$
Work Step by Step
$\int_{e}^{e^4}\frac{dx}{x\sqrt{ln~x}}$
Let $u = ln~x$
$\frac{du}{dx} = \frac{1}{x}$
$dx = x~du$
When $x = e$, then $u = 1$
When $x = e^4$, then $u = 4$
$\int_{1}^{4} \frac{x~du}{x\sqrt{u}}$
$=\int_{1}^{4} \frac{du}{\sqrt{u}}$
$=2\sqrt{u}~\vert_{1}^{4}$
$=2\sqrt{4}-2\sqrt{1}$
$=4-2$
$= 2$