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  "body": "\n## Summary\n\nAn elliptic integral is an integral of a rational function of $t$ and $\\sqrt{P(t)}$.\nHere $P$ is a polynomial of degree $3$ or $4$ without repeated roots.\nSuch integrals arose from the arc length of the [[ellipse|0VKCCEXBJ1]] and in general cannot be written with elementary functions.\nLegendre showed that every elliptic integral reduces to elementary functions and three standard integrals, of the first, second and third kind.\nInverting the integral of the first kind leads to the [[Jacobi elliptic functions|P071B692JF]].\n\n## Explanations\n\n### L1 human\n\nIntegrals with $\\sqrt{1 - t^2}$ give the arcsine and the length of a circle arc.\nReplacing the square root of a quadratic by the square root of a cubic or quartic gives an elliptic integral.\nThe simplest case is the length of an arc of an ellipse, which is the origin of the name.\nFor example, $\\int_0^{x} dt / \\sqrt{(1 - t^2)(1 - k^2 t^2)}$ with $0 \u003c k \u003c 1$ is an elliptic integral; for $k = 0$ it collapses to $\\arcsin x$.\nThese integrals define new functions, which tables and computers evaluate like the logarithm or the sine.\n",
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