Answer
$$\int^e_{1}\frac{\sqrt{\ln x}}{x}dx=\frac{2}{3}$$
Work Step by Step
$$A=\int^e_{1}\frac{\sqrt{\ln x}}{x}dx$$
We set $a=\ln x$, which means $$da=\frac{1}{x}dx$$
For $x=1$, $a=0$ and for $x=e$, $a=1$
Therefore, $$A=\int^{1}_{0} \sqrt ada$$ $$A=\int^{1}_{0} a^{1/2}da$$ $$A=\Big(\frac{a^{3/2}}{\frac{3}{2}}\Big)\Big]^{1}_0=\Big(\frac{2a^{3/2}}{3}\Big)\Big]^{1}_0$$ $$A=\frac{2\times1^{3/2}}{3}-\frac{2\times0^{3/2}}{3}$$ $$A=\frac{2}{3}-0=\frac{2}{3}$$
