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\(\Rightarrow C=1+\left[\frac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(1-\sqrt{x}\right)\left(\sqrt{x}+1\right)}-\frac{2x\sqrt{x}-\sqrt{x}+x}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].\frac{x-\sqrt{x}}{2\sqrt{x}-1}\)
\(=1+\left[\frac{\left(2\sqrt{x}-1\right)\left(1+\sqrt{x}+x\right)-\left(2x\sqrt{x}-\sqrt{x}+x\right)}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].\frac{x-\sqrt{x}}{2\sqrt{x}-1}\)
\(=1+\left[\frac{2\sqrt{x}+2x+2x\sqrt{x}-1-\sqrt{x}-x-2x\sqrt{x}+\sqrt{x}-x}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{2\sqrt{x}-1}\)
\(=1+\left[\frac{2\sqrt{x}-1}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}+x\right)}\right].-\frac{\sqrt{x}\left(1-\sqrt{x}\right)}{2\sqrt{x}-1}\)
\(=1-\frac{\sqrt{x}}{1+\sqrt{x}+x}\) \(=\frac{1+\sqrt{x}+x-\sqrt{x}}{1+\sqrt{x}+x}=\frac{1+x}{1+\sqrt{x}+x}\)
Xét : \(1+2x=1+\frac{\sqrt{3}}{2}=\frac{2+\sqrt{3}}{2}=\frac{4+2\sqrt{3}}{4}=\frac{\left(\sqrt{3}+1\right)^2}{4}\)
\(1-2x=1-\frac{\sqrt{3}}{2}=\frac{2-\sqrt{3}}{2}=\frac{4-2\sqrt{3}}{4}=\frac{\left(\sqrt{3}-1\right)^2}{4}\)
Ta có : \(A=\frac{\frac{\left(\sqrt{3}+1\right)^2}{4}}{1+\sqrt{\left(\frac{\sqrt{3}+1}{2}\right)^2}}+\frac{\frac{\left(\sqrt{3}-1\right)^2}{4}}{1-\sqrt{\left(\frac{\sqrt{3}-1}{2}\right)^2}}\)
\(=\frac{\frac{\left(\sqrt{3}+1\right)^2}{4}}{1+\frac{\sqrt{3}+1}{2}}+\frac{\frac{\left(\sqrt{3}-1\right)^2}{4}}{1-\frac{\sqrt{3}-1}{2}}=\frac{\left(\sqrt{3}+1\right)^2}{2\left(3+\sqrt{3}\right)}+\frac{\left(\sqrt{3}-1\right)^2}{2\left(3-\sqrt{3}\right)}\)
\(=\frac{1}{2\sqrt{3}}\left(\frac{4+2\sqrt{3}}{\sqrt{3}+1}+\frac{4-2\sqrt{3}}{\sqrt{3}-1}\right)=\frac{1}{2\sqrt{3}}.\frac{4\sqrt{3}-4+6-2\sqrt{3}+4\sqrt{3}+4-6-2\sqrt{3}}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)
\(=\frac{1}{2\sqrt{3}}.\frac{4\sqrt{3}}{2}=1\)