ĐKXĐ : \(\left\{{}\begin{matrix}x^3\ge0\\x-1\ge0\\\sqrt{x}-1\ne0\\\sqrt{x-1}\pm\sqrt{x}\ne0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x\ge0\\x\ge1\\x\ne1\\\sqrt{x-1}\ne\pm\sqrt{x}\end{matrix}\right.\)
=> \(\left\{{}\begin{matrix}x>1\\x-1\ne x\end{matrix}\right.\) => \(\left\{{}\begin{matrix}x>1\\0\ne-1\end{matrix}\right.\) => x > 1
Ta có : \(\frac{1}{\sqrt{x-1}-\sqrt{x}}+\frac{1}{\sqrt{x-1}+\sqrt{x}}+\frac{\sqrt{x^3}-x}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x-1}+\sqrt{x}}{\left(\sqrt{x-1}-\sqrt{x}\right)\left(\sqrt{x-1}+\sqrt{x}\right)}+\frac{\sqrt{x-1}-\sqrt{x}}{\left(\sqrt{x-1}+\sqrt{x}\right)\left(\sqrt{x-1}-\sqrt{x}\right)}+\frac{x\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x-1}+\sqrt{x}}{x-1-x}+\frac{\sqrt{x-1}-\sqrt{x}}{x-1-x}+x\)
\(=\frac{\sqrt{x-1}+\sqrt{x}}{-1}+\frac{\sqrt{x-1}-\sqrt{x}}{-1}+x\)
\(=\frac{\sqrt{x-1}+\sqrt{x}+\sqrt{x-1}-\sqrt{x}}{-1}+x\)
\(=\frac{2\sqrt{x-1}}{-1}+x\)
