Answer
$$\frac{1}{2}\pi $$
Work Step by Step
$$\eqalign{
& \int_0^\pi {{{\sin }^2}5\theta } d\theta \cr
& {\text{use the half - angle formula si}}{{\text{n}}^2}x = \frac{1}{2} - \frac{{\cos 2x}}{2}.{\text{ consider }}x = 5\theta \cr
& \int_0^\pi {{{\sin }^2}5\theta } d\theta = \int_0^\pi {\left( {\frac{1}{2} - \frac{{\cos 10\theta }}{2}} \right)} d\theta \cr
& = \int_0^\pi {\frac{1}{2}} d\theta - \int_0^\pi {\frac{{\cos 10\theta }}{2}} d\theta \cr
& = \frac{1}{2}\int_0^\pi {d\theta } - \frac{1}{{2\left( {10} \right)}}\int_0^\pi {\cos 10\theta } \left( {10} \right)d\theta \cr
& {\text{integrate}} \cr
& = \left( {\frac{1}{2}\theta - \frac{1}{{20}}\sin 10\theta } \right)_0^\pi \cr
& {\text{evaluate the limits}} \cr
& = \left( {\frac{1}{2}\left( \pi \right) - \frac{1}{{20}}\sin 10\left( \pi \right)} \right) - \left( {\frac{1}{2}\left( 0 \right) - \frac{1}{{20}}\sin 10\left( 0 \right)} \right) \cr
& {\text{simplify}} \cr
& = \frac{1}{2}\pi \cr} $$
