Ta có: \(P=\left(\frac{1}{\sqrt{x}+2}+\frac{\sqrt{x}}{2-\sqrt{x}}+\frac{2x-\sqrt{x}+2}{x-4}\right):\frac{\sqrt{x}}{x-7}\)
\(=\frac{\sqrt{x}-2-\sqrt{x}\left(\sqrt{x}+2\right)+2x-\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\cdot\frac{x-7}{\sqrt{x}}\)
\(=\frac{2x-x-2\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\cdot\frac{x-7}{\sqrt{x}}=\frac{x-2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-2\right)}\cdot\frac{x-7}{\sqrt{x}+2}\)
\(=\frac{x-7}{\sqrt{x}+2}\)
Để P nguyên thì x-7⋮\(\sqrt{x}+2\)
=>\(x-4-3\) ⋮\(\sqrt{x}+2\)
=>-3⋮\(\sqrt{x}+2\)
=>\(\sqrt{x}+2\in\left\lbrace1;-1;3;-3\right\rbrace\)
=>\(\sqrt{x}=1\)
=>x=1
=>P chỉ đạt giá trị nguyên duy nhất tại x=1
