The Relationship Between a Function, a Functions Inverse, and their Antiderivatives with an Emphasis in Finding Exact Roots with the Technique of Integration

TL;DR

This paper introduces a novel integration-based technique for finding exact roots of smooth functions with inverses and antiderivatives, eliminating the need for numerical or iterative methods.

Contribution

It presents a new analytical method leveraging integration to determine exact roots of specific functions, expanding traditional root-finding approaches.

Findings

Exact roots can be obtained analytically for certain functions.

The technique applies to smooth functions with invertible and integrable inverses.

It bypasses numerical and iterative root-finding methods.

Abstract

Using a new technique involving integration it is possible to find the exact roots of simple functions. In this case, simple functions are defined as smooth functions having an inverse, and that inverse having an antiderivative. This technique now makes it possible to find the exact roots of certain functions without the use of numerical or iterative methods.