$\underset{n\to \infty }{Lt}\sum _{ r=1}^n\frac{1}{n}\sqrt{\frac{n+r}{n-r}}=$

$\underset{n\to \infty }{Lt}\underset{r=1}{\overset{n}{ \sum }}\frac{1}{n}\sqrt{\frac{n+r}{n-r}}$

$=\underset{n\to \infty }{Lt}\frac{1}{n}\sum _{r=1}^n\sqrt{\frac{1+r/n}{1-r/n}}$

$=\underset{0}{\overset{1}{\int }}\sqrt{\frac{1+x}{1-x}}dx$

$=\underset{0}{\overset{1}{\int }}\frac{1+x}{\sqrt{1-x^2}}dx$

$=\underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{1-x^2}}dx+\underset{0}{\overset{1}{\int }}\frac{x}{\sqrt{1-x^2}}dx$

$={\left({\sin }^{-1}x\right)}_0^1+\left(\frac{-1}{2}\right)\underset{0}{\overset{1}{\int }}\frac{-2x}{\sqrt{1-x^2}}dx$

$=({\sin }^{-1}1-{\sin }^{-1}0)-\frac{1}{2}{\left(2\sqrt{1-x^2}\right)}_0^1$

$\begin{array}{l}=\frac{\pi }{2}-(\sqrt{0}-\sqrt{1})\\ =\frac{\pi }{2}+1\end{array}$