Answer
$\dfrac{2 (\ln 2)^{3/2}}{3}$
Work Step by Step
Given: $I=\int_{1}^{2} \dfrac{\sqrt {\ln x}}{x} \ dx$
Let us consider that $u=\ln (x) \implies dx=x \ du$
Now, we have $I=\int_{1}^{2} u^{1/2} \ du$
or, $=[\dfrac{u^{3/2}}{3/2}]_1^2+C$
or, $=[\dfrac{2 u^{3/2}}{3}]_1^2+C$
or, $=[\dfrac{2 (\ln x)^{3/2}}{3}]_1^2+C$
Now, $[\dfrac{2 (\ln (2))^{3/2}}{3}]-[\dfrac{2 (\ln (1))^{3/2}}{3}]=\dfrac{2 (\ln 2)^{3/2}}{3}$
