\[\begin{array}{l} \int\limits_0^{\frac{\pi }{2}} {{{\sin }^{2k + 1}}xdx} = \frac{{\left( {2k} \right)!!}}{{\left( {2k + 1} \right)!!}}, \hfill \\ \int\limits_0^{\frac{\pi }{2}} {{{\sin }^{2k}}xdx} = \frac{{\left( {2k - 1} \right)!!}}{{\left( {2k} \right)!!}} \cdot \frac{\pi }{2}, \hfill \\ k \in \mathbb{N} \hfill \\ \end{array} \]