Fuzzy Nature of a Conjecture on Nonelementary Integrals

Abstract

Fuzziness is also seen in the nature of the integrals of some special type of elementary functions in binary nature i.e. it doesn’t lie between yes and no but it is either yes or no. In this case the fuzziness behaves like a member of a classical set or crisp set and as per mathematical logic it is either true (elementary) or false (nonelementary) i.e. either belongs to the set or doesn’t belong to the set, where the set denotes the collection of all elementary functions. In this paper we have propounded a conjecture on antiderivative containing the functions made of inverse trigonometric functions and polynomials, where a particular function written in the numerator is made up by a composition of two functions, the inverse trigonometric functions and a polynomial function. The integrand contains two polynomials, which may or may not be equal, one as an argument in the inverse trigonometric function and another as a denominator of the integrand. The computing software Mathematica has played a vital role in integrating the complex functions originating as a particular case of the proposed conjecture. It has been found that the nature of the functions written as integrands varies as we increase the degree of the polynomials written both in numerator and denominator. Two very interesting integrals have been found in the study, which are always elementary containing inverse tangent and cotangent functions. The paper ends with the conclusion, its limitations and the scope of further research.