$\displaystyle \begin{array}{{>{\displaystyle}l}} P=2\left(\frac{1}{\sqrt{x-1}} -\frac{1}{\sqrt{x-1} +1}\right) :\frac{\sqrt{x-1}}{x+\sqrt{x-1} -1} \ \\ P=2.\frac{\sqrt{x-1} +1-\sqrt{x-1}}{\left(\sqrt{x-1}\right)\left(\sqrt{x-1} +1\right)} .\frac{x+\sqrt{x-1} -1}{\sqrt{x-1}} \ \\ P=\frac{x+\sqrt{x-1} -1}{( x-1)\left(\sqrt{x-1} +1\right)} =\frac{\left(\sqrt{x-1}\right)\left(\sqrt{x-1} +1\right)}{( x-1)\left(\sqrt{x-1} +1\right)} =\frac{1}{\sqrt{x-1}} \ \\ Để\ P\ nguyên\ thì\ \frac{1}{\sqrt{x-1}} \ nguyên\ \\ \rightarrow để\ P\ nguyên\ thì\ \sqrt{x-1} \ là\ số\ nguyên\ khi\ \sqrt{x-1} =1\ \rightarrow \ x=2\ \\ Hay\ \sqrt{x-1} =\frac{1}{k\ } \ ( \ k\ nguyên,k\#0\ ) \ \\ \rightarrow \ x-1=\frac{1}{k^{2}}\rightarrow \ x=\frac{1}{k^{2}} +1\ \\ \\ \end{array}$
