Some Applications of Elliptic Integrals of First Kind

One of the most important applications of elliptic integrals in engineering mathematics is their usage to solve integrals of the form ∫ 𝑅(𝑥, √𝑃𝑛(𝑥))𝑑𝑥 (Eq. 1), where 𝑅 is a rational algebraic function and 𝑃𝑛(𝑥) is a polynomial of degree 𝑛 = 3,4 with no repeated roots. Nowadays, incomplete and complete elliptic integrals of first kind are estimated with high accuracy using advanced calculators. In this paper, several techniques are discussed to show how definite integrals of the form (Eq. 1) can be converted to elliptic integrals of the first kind, and hence be estimated for optimal values. Indeed, related examples are provided in each step to help clarification.