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$$\int\limits_a^bp(x)f(x)\,dx\approx\sum_{i=1}^nc_if(x_i),$$

in which the nodes (cf. Node) $x_i$ and the weights $c_i$ are so selected that the formula is exact for the functions

$$\sum_{k=0}^{2n-1}a_k\omega_k(x),$$

where $\omega_k(x)$ are given linearly independent functions (the integration limits may well be infinite). The formula was introduced by C.F. Gauss  for $a=-1$, $b=1$, $p(x)\equiv1$. He obtained the following formula, which is exact for an arbitrary polynomial of degree not exceeding $2n-1$:

$$\int\limits_{-1}^1f(x)\,dx=A_1^{(n)}f(x_1)+\dots+A_n^{(n)}f(x_n)+R_n,$$

where the $x_k$ are the roots of the Legendre polynomial (cf. Legendre polynomials) $P_n(x)$, while $A_k^{(n)}$ and $R_n$ are defined by the formulas

$$A_k^{(n)}=\frac{2}{(1-x_k^2)[P_n'(x_k)]^2};$$

$$R_n=\frac{2^{2n+1}[n!]^4}{(2n+1)[(2n)!]^3}f^{(2n)}(c),\qquad-1<c<1.$$

The formula is used whenever the integrand is sufficiently smooth, and the gain in the number of nodes is substantial; for instance, if $f(x)$ is determined from expensive experiments or during the computation of multiple integrals as repeated integrals. In such practical applications a suitable choice of the weight function and of the functions $\omega_j(x)$ is very important.

Tables of nodes in Gauss' quadrature formula are available for wide classes of $p(x)$ and $\omega_j(x)$ ; in particular for $p(x)\equiv1$, $\omega_j(x)=x^j$ up to $n=512$.

If $p(x)\equiv1$, $\omega_j(x)=x^j$, Gauss' quadrature formula is employed in standard integration programs with an automatic step selection as a method of computing integrals by subdivision of subsegments .

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