Answer
$$\frac{1}{2}\theta - \frac{1}{{20}}\sin 10\theta + C$$
Work Step by Step
$$\eqalign{
& \int {{{\sin }^2}5\theta } d\theta \cr
& {\text{identity si}}{{\text{n}}^2}x = \frac{{1 - \cos 2x}}{2},{\text{ }}x = 5\theta \cr
& = \int {\frac{{1 - \cos 2\left( {5\theta } \right)}}{2}} d\theta \cr
& = \int {\frac{{1 - \cos 10\theta }}{2}} d\theta \cr
& = \int {\left( {\frac{1}{2} - \frac{{\cos 10\theta }}{2}} \right)} d\theta \cr
& {\text{sum rule}} \cr
& = \int {\frac{1}{2}} d\theta - \frac{1}{2}\int {\cos 10\theta } d\theta \cr
& {\text{find antiderivatives}} \cr
& = \frac{1}{2}\theta - \frac{1}{2}\left( {\frac{1}{{10}}\sin 10\theta } \right) + C \cr
& = \frac{1}{2}\theta - \frac{1}{{20}}\sin 10\theta + C \cr} $$
