I've been refreshing my numerical analysis and reading about quadrature formulas, that is, how to construct algorithms that estimate the value of a definite integral. I know that, in general, is not a great idea to choose equidistant nodes. In fact, some of the best quadrature formulas (for example, Chebyshev quadrature formula) use nodes that are more densely distributed near the endpoints of the interval on which we want to estimate the integral.
My question is: why? Could you explain the main idea behind this?
The only place where equidistant nodes degrade the accuracy of quadrature is in high-order Newton–Cotes formulas, the set of quadrature rules that generalises the trapezoidal, Simpson's and Boole's rules. All these rules fit a polynomial to equidistant points, but as the degree goes up Runge's phenomenon kicks in and induces wild oscillations in the interpolating polynomial for many smooth inputs.
There are two main ways to avoid this problem: using nodes clustered around the integration endpoints as you noted (Gauss–Legendre is under this case too), or subdividing the integration interval and using a low-order rule on each little piece (i.e. "spline" integration).