Overview

"The usefulness and uniqueness of exponential functions in the world of mathematics and science has been proven imperative over the years. However, It is common experience for 'all users of mathematics' who have encountered integrals of exponential functions in these forms, ∫xⁿe^(ax)dx and ∫xⁿe^(-ax)dx, to observe that the resolution of those integrals follow recursive patterns and sometimes seem interminable especially when 'n' is relatively large, meanwhile, the latter statement is not true as we will see later in the text. Most often, especially when 'n' is moderately large in ∫xⁿe^(ax)dx or ∫xⁿe^(-ax)dx, students / teachers of mathematics who come across integrals of those forms above, either shy away from resolving them, or they get bored on the process. Hence, the need to improve on the reduction formula(process) or ameliorate the use of integration by parts in resolving and utilizing the 'inevitable integrals' informed the idea in this paper titled ""PURGE THE RECURSIVE"". Consequently, G.C.F.R's integral representations - is a book that proffer compact integral representations for integrals of these forms, ∫xⁿe^(ax)dx and ∫xⁿe^(-ax)dx. Alongside, the book aims at acquainting 'all users of mathematics' with a 'stress-free' approach to resolving definite and indefinite integrals of those forms stated above, without having to work with the usual reduction formula or integration by parts. This approach is significantly efficient over the reduction formula or integration by parts as it precludes protracted complexities that are predominant when 'n' in the integrals stated above is moderately large. Furthermore, the paper provides substantial analytical proofs for major Gamma properties(function) via the utilization of G.C.F.R's analytical approach. Also, Analogous to the celebrated Gamma function, G.C.F.R's integral representations which are prime to this book, seek to enhance continuity in infinitesimal calculus by linking two disparate branches of mathematics, that is, Number theorem and Calculus via its wide-ranging representations. The study maximizes the method of integration by parts in deducing the integrals of some randomly chosen integrands. G.C.F.R's integral representations were obtained via principle of mathematical induction."

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