Answer
$$\pi $$
Work Step by Step
$$\eqalign{
& \int_{ - \pi }^\pi {{{\cos }^2}5\theta } d\theta \cr
& {\text{identity }}{\cos ^2}x = \frac{{1 + \cos 2x}}{2} \cr
& = \int_{ - \pi }^\pi {\frac{{1 + \cos 2\left( {5\theta } \right)}}{2}} d\theta \cr
& = \int_{ - \pi }^\pi {\frac{{1 + \cos 2\left( {5\theta } \right)}}{2}} d\theta \cr
& = \int_{ - \pi }^\pi {\left( {\frac{1}{2} + \frac{{\cos 10\theta }}{2}} \right)} d\theta \cr
& {\text{find antiderivative}} \cr
& = \left[ {\frac{\theta }{2} + \frac{{\sin 10\theta }}{{20}}} \right]_{ - \pi }^\pi \cr
& {\text{fundamental theorem of calculus}} \cr
& = \left[ {\frac{\pi }{2} + \frac{{\sin 10\pi }}{{20}}} \right] - \left[ {\frac{{ - \pi }}{2} - \frac{{\sin 10\pi }}{{20}}} \right] \cr
& {\text{simplifying}} \cr
& = \left[ {\frac{\pi }{2} + 0} \right] - \left[ {\frac{{ - \pi }}{2} - 0} \right] \cr
& = \frac{\pi }{2} + \frac{\pi }{2} \cr
& = \pi \cr} $$
