Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of $\int_{a}^{b}f(x)dx$.

I can also see that the converse does not necessarily hold. That is,

If one application of the trapezoidal rule over $a\leq x \leq b$ yields the exact value of $\int_{a}^{b}f(x)dx$, then it is not necessarily the case that $f(x)$ has constant gradient.

Any function that is odd about the inflexion point would act as counter examples to demonstrate this.

My question is, are such functions the ONLY non-linear functions whose integrals over $a\leq x \leq b$ can be found using a single trapezium over the same domain?