$ = \int_0^{\pi /2} {\frac{{dx}}{{2{{\sin }^2}\frac{x}{2} + 2{{\cos }^2}\frac{x}{2} + {{\cos }^2}\frac{x}{2} - {{\sin }^2}\frac{x}{2}}}} $
$ = \int_0^{\pi /2} {\frac{{dx}}{{{{\sin }^2}\frac{x}{2} + 3{{\cos }^2}\frac{x}{2}}}} $
$= \int_0^{\pi /2} {\frac{{{{\sec }^2}\frac{x}{2}}}{{3 + {{\tan }^2}\frac{x}{2}}}dx} $
Put $t = \tan \frac{x}{2} $
$\Rightarrow dt = \frac{1}{2}{\sec ^2}\frac{x}{2}dx$, then
$I = 2\int_0^1 {\frac{{dt}}{{3 + {t^2}}} = \frac{2}{{\sqrt 3 }}{{\tan }^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)} $.
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