= Symmetric trapezoidal exactness for odd functions
{title2=$T_N(g)=\int_{-L}^Lg(x)\,dx=0\quad(g\text{ odd})$}
On a <partition of an interval> symmetric about zero, the <composite trapezoidal rule> gives opposite <quadrature nodes> equal <quadrature weights>. Samples of an <odd function> cancel pairwise, and a zero <quadrature node> contributes zero. For a continuous <odd function>, its exact <integral> over $[-L,L]$ is also zero. Adding any <affine function> preserves exactness by <linearity of the integral> and of the <quadrature rule>. For example, $1+\sin x$ is nonpolynomial but both the exact <integral> and every uniform-grid <composite trapezoidal rule> value on $[-1,1]$ are two.
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