A simple harmonic oscillator has amplitude A and force constant k. Its average potentíal energy over one time period T is

The potential energy of the oscillator an time t is given by
$U=\frac{1}{2}k{x}^2=\frac{1}{2}k{A}^2{\sin }^2(\omega t+\varphi )$
The average potential energy from t=0 to t=T is
$$\begin{gathered} U_{av}=\frac{{\int }_0^{T}Udt}{{\int }_0^{T}dt}=\frac{1}{T}{\int }_0^T\frac{1}{2}k{A}^2{\sin }^2(\omega t+\varphi )dt \\ =\frac{k{A}^2}{2T}{\int }_0^T{\sin }^2(\omega t+\varphi )dt \\ =\frac{k{A}^2}{4T}{\int }_0^T[1-\cos (2\omega t+\varphi \left)\right]dt \\ =\frac{k{A}^2}{4T}(T-0)=\frac{1}{4}k{A}^2 \end{gathered}$$