Para encontrarmos aintegral da função dada, realizaremos os cálculos abaixo:

\(\begin{align} & f(x)=\sqrt{({{(cos(x))}^{2}}+1)} \\ & \\ & \int_{{}}^{{}}{f(x)dx}=\int_{0}^{2\pi }{\sqrt{({{(cos(x))}^{2}}+1)}} \\ & \int_{{}}^{{}}{f(x)dx}=\int_{0}^{2\pi }{\sqrt{\frac{\cos 2x+3}{2}}} \\ & \int_{{}}^{{}}{f(x)dx}=\int_{0}^{2\pi }{\frac{\sqrt{\cos 2x+3}}{\sqrt{2}}} \\ & \int_{{}}^{{}}{f(x)dx}=\left[ \sqrt{2}E\left( x|\frac{1}{2} \right) \right]_{0}^{2\pi } \\ & \int_{{}}^{{}}{f(x)dx}=2\pi n+\frac{\pi }{2} \\ \end{align}\ \)

Portanto, aintegral será \(\begin{align} & \int_{{}}^{{}}{f(x)dx}=2\pi n+\frac{\pi }{2} \\ \end{align}\ \).