eng Ansari Education and Research Society Journal of Ultra Scientist of Physical Sciences october 2026 38 1 1 11 http://dx.doi.org/10.22147/jusps-A/380101 1588 article JANHABI BHUYAN 1 SUNITA KUMARI NAYAK 1 Department of Mathematics and Applied Statistics, School of Applied Sciences Kalinga Institute of Industrial Technology (KIIT) Deemed to be University, Bhubaneswar - 751024, Odisha, India <p style="text-align:justify">This paper proposes an efficient quadrature rule of precision eleven for the numerical evaluation of improper integrals with endpoint singularities. The proposed method combines Gaussian and Newton&ndash;Cotes quadrature rules of lower precision within a unified framework, incorporating cubic Newton divided differences and Romberg extrapolation to enhance computational efficiency and accuracy. An error bound is derived using a mixed quadrature approach, providing a theoretical basis for the performance of the proposed scheme. Numerical experiments and convergence analyses demonstrate its accuracy, stability, and efficiency. The proposed method is further validated through comparisons with existing quadrature techniques and available analytical solutions. Moreover, the Mellin transform is employed to accelerate the numerical computations and improve the accuracy of the resulting approximations. The results indicate that the proposed efficient quadrature scheme provides an effective and reliable approach for the numerical treatment of singular improper integrals</p> https://www.ultrascientist.org/paper/1588/ efficient quadrature Mellin transform cubic Newton divided differences