$ = \int {\frac{{{{\sec }^2}(x/2)\,dx}}{{7 + 7{{\tan }^2}(x/2) + 5 - 5{{\tan }^2}(x/2)}}} $
$ = \int {\frac{{{{\sec }^2}(x/2)\,dx}}{{12 + 2{{\tan }^2}(x/2)}}} $$ = \int {\frac{{\frac{1}{2}{{\sec }^2}(x/2)\,.dx}}{{6 + {{\tan }^2}(x/2)}}} $
$\tan \frac{x}{2} = t$ $⇒$ $\frac{1}{2}{\sec ^2}\frac{x}{2}dx = dt$
$I = \int {\frac{{dt}}{{{t^2} + ({{\sqrt {6)} }^2}}}} $ $ = \frac{1}{{\sqrt 6 }}{\tan ^{ - 1}}\frac{t}{{\sqrt 6 }} + c$
$ = \frac{1}{{\sqrt 6 }}{\tan ^{ - 1}}\left| {\frac{{\tan (x/2)}}{{\sqrt 6 }}} \right| + c$
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$\begin{array}{|l|l|l|l|l|l|} \hline X=x & 0 & 1 & 2 & 3 & 4 \\ \hline P(X=x) & \frac{1}{3} & \frac{1}{2} & 0 & \frac{1}{6} & 0 \\ \hline \end{array}$
, then
$\mathrm{g}^{\prime \prime}\left(\mathrm{N}+\frac{1}{2}\right)-\mathrm{g}^{\prime \prime}\left(\frac{1}{2}\right)=$