[tex]\\ \begin{array}{rcl} \displaystyle \int \dfrac{\ln x}{\sqrt{x}} \, dx&=& \displaystyle \int (2\sqrt x)'\cdot \ln x \, dx \\ &=& \displaystyle 2\sqrt x \cdot \ln x - \int 2\sqrt x \cdot (\ln x)' \, dx \\ &=& \displaystyle 2\sqrt{x} \cdot \ln x -2 \int \sqrt x \cdot \dfrac{1}{x} \, dx \\ &=& \displaystyle 2\sqrt x\cdot \ln x - 2\int \dfrac{1}{\sqrt x} \, dx \\ &=& \displaystyle 2\sqrt x\cdot \ln x - 2\int (2\sqrt x)' \, dx \\ &=& 2\sqrt x\cdot \ln x - 2\cdot 2\sqrt x + C \\ \\ &=& \boxed{2\sqrt x\cdot (\ln x - 2) + C}\end{array}[/tex]
