$\underset{2}{\overset{5}{\int }}\sqrt{\frac{5-x}{x-2}}dx=$

$\begin{array}{l}Put x-2=3{\sin }^2\theta \Rightarrow dx=3\times 2\sin \theta \cos \theta d\theta \\ L.L\Rightarrow x=2\Rightarrow \theta =0\\ U.L\Rightarrow x=5\to 5-2=3{\sin }^2\theta \\ \theta =\frac{\pi }{2}\end{array}$

 $\underset{2}{\overset{5}{\int }}\sqrt{\frac{5-x}{x-2}}dx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sqrt{\frac{3{\cos }^2\theta }{3{\sin }^2\theta }}3\sin 2\theta d\theta$

$=3\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\cos \theta }{\sin \theta }2\sin \theta \cos \theta d\theta$

$=6\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^2\theta d\theta$

$=\frac{6}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}(1+\cos 2\theta )d\theta =3{[\theta +\frac{\sin 2\theta }{2}]}_0^{\frac{\pi }{2}}=3\left[\frac{\pi }{2}\right]=\frac{3\pi }{2}$