Question

$\int {\left(\frac{\sqrt{x}-}{2}\frac{1}{\sqrt{x}}\right)}^2(\frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{\sqrt{x}+1}{\sqrt{x}-1})dx,x>0$

Answer 1

$\begin{array}{c}\int {\left(\frac{\sqrt{x}-1}{2\sqrt{x}}\right)}^2\left[\frac{\sqrt{x}-1}{\sqrt{x}+1}\times \frac{\sqrt{x}-1}{\sqrt{x}-1}-\frac{\left(\sqrt{x}-1\right)\times \left(x+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right]\\ \Rightarrow \frac{1}{4}\int {\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)}^2\left[\frac{{\left(\sqrt{x}-1\right)}^2}{\sqrt{x}-1}-\frac{{\left(\sqrt{x}+1\right)}^2}{\left(x-1\right)}\right]\\ \Rightarrow \frac{1}{4}\int \left[\frac{{\left(x-1\right)}^2}{x^2}\right]\left[\frac{2\sqrt{x}\times \left(-2\right)}{\left(x-1\right)}\right]\\ \Rightarrow \frac{-4}{4}\int \frac{\left(x-1\right)}{\sqrt{x}}dx\\ \Rightarrow -1\int \sqrt{x}dx+\int x^{\frac{-1}{2}}dx\\ \Rightarrow -\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+c\\ \Rightarrow -\frac{2}{3}{x}^{\frac{3}{2}}+2x^{\frac{1}{2}}+c\\ 2\left[\sqrt{x}-\frac{{\left(\sqrt{x}\right)}^3}{3}\right]+c\end{array}$