Answer
$$\int_e^{e^4}\frac{dx}{x\sqrt{\ln x}}=2$$
Work Step by Step
To evaluate the integral
$$\int_e^{e^4}\frac{dx}{x\sqrt{\ln x}}$$
we will use substitution $\ln x=t$ which gives us $\frac{dx}{x}=dt$. The integration bounds would be: for $x=e$ we have $t=1$ and for $x=e^4$ we have $t=4$. Putting this into the integral we get:
$$\int_e^{e^4}\frac{dx}{x\sqrt{\ln x}}=\int_1^4\frac{1}{\sqrt t}dt=\int_1^4t^{-1/2}dt=\left.\frac{t^{1/2}}{\frac{1}{2}}\right|_1^4=2(4^{1/2}-1^{1/2})=2(2-1)=2$$
