elliptic integral

Try to compute the exact length of an arc of an ellipse, or the period of a pendulum swinging through a large angle, and you bump into an integral that simply has no antiderivative among the elementary functions. These are elliptic integrals — integrals of a rational function of x and the square root of a cubic or quartic polynomial in x. The name comes from the very first such problem historically: the arc length of an ellipse. They cannot be done with logs, exponentials, and trig functions, no matter how clever you are.

Mathematicians did not give up; they gave these integrals names and tabulated them, just as earlier generations had done for the logarithm and the sine. After reduction to standard forms (the Legendre forms), every elliptic integral is built from three basic non-elementary functions: the elliptic integral of the first kind F, of the second kind E, and of the third kind. The pendulum period, for instance, is exactly a complete elliptic integral of the first kind in the swing amplitude — an exact, computable answer, just not an elementary formula. Inverting the first-kind integral gives the Jacobi elliptic functions, the genuine periodic 'trig functions' of this richer world.

The crucial honesty here, and the whole point of teaching it as a named object, is this: non-elementary does not mean unsolved, unknown, or uncomputable. Elliptic integrals are completely understood, rapidly computable to any precision, and as concrete as the sine — they simply are not expressible in the small alphabet of elementary functions. Treating them as legitimate functions in their own right is exactly how applied mathematics handles the limits of elementary integration honestly rather than pretending the integral does not exist.