The value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{{\cos }^{2}x}{1+3^x}dx$ is 

$\begin{array}{rcl}\mathrm{I}& =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{{\cos }^2\mathrm{x}}{1+3^{\mathrm{x}}} \mathrm{dx}\\ & =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{{\cos }^2(-\mathrm{x})}{7+3^{-\mathrm{x}}} \mathrm{dx}\\ \mathrm{I}& =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{3^{\mathrm{x}} {\cos }^2\mathrm{x}}{3^{\mathrm{x}}+1} \mathrm{dx}\\ 2\mathrm{I}& =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{{\cos }^2\mathrm{x} (1+3^{\mathrm{x}})}{3^{\mathrm{x}}+1} \mathrm{dx}\\ 2\mathrm{I}& =& {\int }_{\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \frac{{\cos }^2\mathrm{x} (1+3^{\mathrm{x}})}{3^{\mathrm{x}}+1} \mathrm{dx}\\ 2\mathrm{I}& =& 2{\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} {\cos }^2\mathrm{x} \mathrm{dx}\\ \mathrm{I}& =& \frac{1}{2}.\frac{\mathrm{\pi }}{2}=\frac{\mathrm{\pi }}{4}\end{array}$