 Wavevector distribution of an infinite-well ground state

A centred width-$L$ infinite square well has ground state wavefunction $\sqrt{2/L}\cos (\pi x/L)$ inside and zero outside. Its momentum representation in quantum mechanics follows from the Fourier transform of a truncated cosine. The probability density in the title has removable singularities at $k=\pm \pi /L$, with limit $L/(4\pi )$, and an oscillatory $k^{-4}$ tail envelope. The distribution is even, with $⟨k⟩=0$ and $⟨k^2⟩={\pi }^2/L^2$.