The methods we have presented hitherto are tailored to problems where the mesh points $x_i$ are equidistantly spaced, $x_i$ differing from $x_{i+1}$ by the step $h$.

The basic idea behind all integration methods is to approximate the integral $$\begin{equation*} I=\int_a^bf(x)dx \approx \sum_{i=1}^N\omega_if(x_i), \end{equation*}$$ where $\omega$ and $x$ are the weights and the chosen mesh points, respectively. In our previous discussion, these mesh points were fixed at the beginning, by choosing a given number of points $N$. The weigths $\omega$ resulted then from the integration method we applied. Simpson's rule, see Eq. (6) would give $$\begin{equation*} \omega : \left\{h/3,4h/3,2h/3,4h/3,\dots,4h/3,h/3\right\}, \end{equation*}$$ for the weights, while the trapezoidal rule resulted in $$\begin{equation*} \omega : \left\{h/2,h,h,\dots,h,h/2\right\}. \end{equation*}$$