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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$.\nLet $x_1, \\dots, x_n$ be the zeros of $P_n$; they are distinct and lie in $(-1, 1)$.\nThe $n$-point *Gauss–Legendre quadrature* rule is\n$$\\int_{-1}^{1} f(x)\\,dx \\approx \\sum_{i=1}^{n} w_i f(x_i), \\qquad w_i = \\frac{2}{(1 - x_i^2)\\, P_n'(x_i)^2}.$$\nIt is the $n$-point [[Gaussian quadrature|FP37BTP93K]] rule for the weight $w = 1$ on $(-1, 1)$.\n\n## Remarks\n\nThe rule is exact for every polynomial of degree at most $2n - 1$.\nFor an interval $[a, b]$ one uses $\\int_a^b g(t)\\,dt = \\frac{b - a}{2} \\int_{-1}^{1} g\\big(\\frac{b - a}{2} x + \\frac{a + b}{2}\\big)\\,dx$.\n\nExample: $n = 2$.\nHere $P_2(x) = \\frac12 (3x^2 - 1)$, with zeros $\\pm 1/\\sqrt{3}$, and $P_2'(x) = 3x$.\nThe weights are $2 / \\big((2/3) \\cdot 3\\big) = 1$.\nFor $f(x) = x^2$ the rule gives $2/3$, and for $f(x) = x^3$ it gives $0$, both exact.\n\nNon-example: for $n = 2$ and $f(x) = x^4$ the rule gives $2/9$ while the integral is $2/5$.\nThis is expected, since $4 \u003e 2n - 1 = 3$.\n",
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