$1. \ \dfrac{4\sqrt{x}+1}{\sqrt{x}+1}=\dfrac{4(\sqrt{x}+1)-3}{\sqrt{x}+1}=4-\dfrac{3}{\sqrt{x}+1}$
$\dfrac{4\sqrt{x}+1}{\sqrt{x}+1}$ nguyên $\Leftrightarrow \dfrac{3}{\sqrt{x}+1}$ nguyên
$\Rightarrow \sqrt{x}+1\in Ư(3)=\left \{ \pm 1;\pm 3 \right \}$
$\Rightarrow x\in \left \{ ........ \right \}$