Elliptic integrals are a class of integrals that arise in the calculation of the arc length of an ellipse. They are defined as integrals of the form ∫ R(√(P(x)), x) dx, where R is a rational function and P is a polynomial of degree three or four. These integrals cannot be expressed in terms of elementary functions, making them important in various fields of mathematics and physics. There are three main types of elliptic integrals: the first kind, the second kind, and the third kind. Each type has specific applications, such as in the study of ellipses, pendulum motion, and complex analysis. Elliptic integrals also have connections to elliptic functions, which are periodic functions that can be expressed in terms of elliptic integrals.