The paper discusses the numerical solution of nonlinear integral equations, focusing on the iterative methods for solving finite systems of nonlinear algebraic or transcendental equations. The author reviews conventional techniques and identifies their limitations, particularly in the context of a specific class of nonlinear equations. A new procedure is proposed to address these disadvantages, which is not restricted to this particular class of systems. The paper is divided into several sections: 1. **Introduction**: Outlines the importance of nonlinear integral equations and the motivation for the proposed methods, based on research in kinetic theory of gases. 2. **The Class of Nonlinear Equations**: Describes the class of nonlinear equations and the discretization process, including the use of Gaussian quadrature and Chebyshev polynomials. 3. **Survey of Conventional Iterative Procedures**: Reviews various iterative methods such as relaxation, Newton-Raphson, and higher-order methods, highlighting their limitations and inefficiencies. 4. **Low-Degree Generalized Secant Methods**: Introduces a heuristic approach using low-degree generalized secant methods, including the extrapolation and relaxation algorithms, and their variants. 5. **Examples**: Provides examples to illustrate the effectiveness of the proposed methods in solving simple and more complex problems. 6. **Conclusion**: Summarizes the advantages of the proposed iterative procedures and their potential applications in a broader context. The paper aims to provide a robust and efficient method for solving nonlinear integral equations, particularly those arising from kinetic theory of gases, by addressing the limitations of existing techniques.The paper discusses the numerical solution of nonlinear integral equations, focusing on the iterative methods for solving finite systems of nonlinear algebraic or transcendental equations. The author reviews conventional techniques and identifies their limitations, particularly in the context of a specific class of nonlinear equations. A new procedure is proposed to address these disadvantages, which is not restricted to this particular class of systems. The paper is divided into several sections: 1. **Introduction**: Outlines the importance of nonlinear integral equations and the motivation for the proposed methods, based on research in kinetic theory of gases. 2. **The Class of Nonlinear Equations**: Describes the class of nonlinear equations and the discretization process, including the use of Gaussian quadrature and Chebyshev polynomials. 3. **Survey of Conventional Iterative Procedures**: Reviews various iterative methods such as relaxation, Newton-Raphson, and higher-order methods, highlighting their limitations and inefficiencies. 4. **Low-Degree Generalized Secant Methods**: Introduces a heuristic approach using low-degree generalized secant methods, including the extrapolation and relaxation algorithms, and their variants. 5. **Examples**: Provides examples to illustrate the effectiveness of the proposed methods in solving simple and more complex problems. 6. **Conclusion**: Summarizes the advantages of the proposed iterative procedures and their potential applications in a broader context. The paper aims to provide a robust and efficient method for solving nonlinear integral equations, particularly those arising from kinetic theory of gases, by addressing the limitations of existing techniques.