 Integer-frequency endpoint expansion of cosine integrals

For smooth $h$ on $[0,2\pi ]$ and positive integer $n$, repeatedly integrating $\int h(t)\cos (nt)dt$ twice eliminates the sine endpoint terms and yields only even inverse powers of $n$. The exact remainder after $M$ pairs is $(-1{)}^M{n}^{-2M}\int h^{(2M)}\cos (nt)$, which is $o(n^{-2M})$ by the Riemann-Lebesgue lemma. Without integer frequencies an $n^{-1}$ sine endpoint term generally remains.