$\int_{\pi /3}^{\pi /2} {\frac{{\sqrt {1 + \cos x} }}{{{{(1 - \cos x)}^{5/2}}}}} dx$

Let $I = \int_{\pi /3}^{\pi /2} {\frac{{\sqrt {1 + \cos x} }}{{{{(1 - \cos x)}^{5/2}}}}} dx$

$= \int_{\pi /3}^{\pi /2} {\frac{{\sqrt {1 + \cos x} }}{{{{(1 - \cos x)}^2}\sqrt {1 + \cos x} }}} dx$

$= \int_{\pi 3}^{\pi 2} {\frac{1}{{\left( {1 - {{\cos }^2}x} \right)}}} dx = \int_{\pi 3}^{\pi 2} {\frac{1}{{{{\sin }^2}x}}} dx$

$= \int_{\pi 3}^{\pi 2} {{{{\mathop{\rm cosec}\nolimits} }^2}} xdx = [ - \cot x]_{\pi /3}^{\pi /2}$

$= - \left[ {\cot \frac{\pi }{2} - \cot \frac{\pi }{3}} \right] = - \left[ {0 - \frac{1}{{\sqrt 3 }}} \right] = + \frac{1}{{\sqrt 3 }}$