Binary Nature of a Conjecture on Nonelementary Integrals in Context of Elementary Functions

Abstract

Binary naturehas beenobserved in the characters of the indefinite integrals of some
elementary functionshaving inverse hyperbolic functions as a component in the
integrand.It doesn’t lie between one (elementary) and zero (nonelementary) but it is
exactly either one or zero. In this case the twocharactersbehave like a member of a classical
set (or crisp set) and in the language of mathematical logic it is either elementary or
nonelementary functions. In this paper we have proffered a conjecture on the
antiderivative containing the elementary functions made of inverse hyperbolicfunctions
and polynomials, where theelementary function written in the numerator is made by
taking the composition of two functions:the inverse hyperbolicfunctions and the
polynomial functions. It contains two polynomials, which may or may not be equal, where
one polynomial occurs as an argument in the inverse hyperbolic functionand another one
as a denominator of the integrand,which means that the integrand is always a fraction. The
computer software mathematica has played animportant role in integrating the assumed
functions originating as a particular case of the proffered conjecture. It has been found that
the nature of the elementary functions written as integrands varies as the degree of the
polynomials increases. Two interesting antiderivatives have been observed in the study,
which are always elementary containing inverse hyperbolic tangent and inverse hyperbolic
cotangent functions. We have ended the paper with conclusion, its limitations and the
further scope of research.