An $n$-point Gaussian quadrature rule on $[a, b]$ with respect to a weight function $w(x) > 0$ is an approximation $\int_a^b w(x) f(x)\, dx \approx \sum_{i=1}^n w_i f(x_i)$ where the nodes $x_1, \ldots, x_n$ and weights $w_1, \ldots, w_n$ are chosen so that the formula is exact for all polynomials of degree $\leq 2n - 1$. This gives the maximum possible degree of exactness for $n$ function evaluations.

For $w(x) = 1$ on $[-1, 1]$, the nodes $x_1, \ldots, x_n$ are the roots of the Legendre polynomial $P_n(x)$, and the weights are $w_i = \frac{2}{(1 - x_i^2)[P_n'(x_i)]^2}$. For a general interval $[a, b]$, the substitution $x = \frac{b-a}{2}t + \frac{a+b}{2}$ transforms the integral: $\int_a^b f(x)\, dx = \frac{b-a}{2} \int_{-1}^1 f\!\left(\frac{b-a}{2}t + \frac{a+b}{2}\right) dt$.

Gauss-Chebyshev quadrature uses $w(x) = \frac{1}{\sqrt{1 - x^2}}$ on $(-1, 1)$, with nodes at $x_k = \cos\!\left(\frac{2k - 1}{2n}\pi\right)$ and equal weights $w_k = \frac{\pi}{n}$. Gauss-Laguerre quadrature uses $w(x) = e^{-x}$ on $[0, \infty)$, with nodes at the roots of Laguerre polynomials $L_n(x)$ and weights $w_i = \frac{x_i}{(n+1)^2 [L_{n+1}(x_i)]^2}$.