Past exam of the numerical analysis course of the University of Oxford / 2015 / 1 / c

On a partition of an interval $a=x_0<x_1<\cdots <x_N=b$, apply the trapezoidal rule separately to each panel and add:For uniform spacing $h=(b-a)/N$, this composite trapezoidal rule becomesEach interior sample occurs in two panels; their half-quadrature weights combine to $h$. The endpoint quadrature weights are $h/2$.

Use the nonpolynomial function $f(x)=1+\sin x$ on $[-1,1]$. Its exact integral isbecause the sine is an odd function.

For every uniform partition of an interval with $N\ge 1$ panels, the quadrature nodes satisfy $x_{N-j}=-x_j$. Their composite trapezoidal rule quadrature weights agree, so the two sampled sine values cancel. If there is a node at zero its sine value is already zero. The constant term is integrated exactly, givingThis is symmetric trapezoidal exactness for odd functions, applied after adding a constant function so that the resulting definite integral is nonzero. It is exactness for this particular integrand, rather than a claim that every nonpolynomial function is integrated exactly.