We consider the trapezoidal rule on $I = [a,b]$ with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$, where $f$ is of class $\mathcal{C}^2$.
Deduce the error bound
$$\left|e_n(f)\right| \leqslant \frac{(b-a)^3}{12n^2} \sup_{x \in [a,b]} |f''(x)|.$$