Estimating the error term in the Trapezium Rule using a Runge-Kutta method

We show how the error term for the Trapezium Rule can be estimated, by solving an initial value problem using a Runge-Kutta method. The error term can then be added to the Trapezium approximation, yielding a much more accurate result. We also show how the risk of singularities in the relevant initial value problem can be mitigated.