Integral evaluation through the binomial formula and differentiation techniques

Abstract

In the realm of mathematics, the evaluation of integrals plays a crucial role in various fields such as physics, engineering, and economics. Integrals represent the accumulation of quantities over a continuous range and are essential for understanding the behavior of functions and solving complex problems. Over the years, mathematicians and researchers have developed numerous techniques to evaluate integrals efficiently and accurately. This thesis consists of two parts. In the first part, we present some binomial identities, special numbers and polynomials as well as basic formulas related to Euler’s transformation of series, Handamard’s series multiplication theorem and several transformation formulas with example and applications. These theoretical tools will be used to gain a better understanding of the second part of this thesis which explores two different techniques for evaluating integrals. The first technique uses a special formula to transform integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integral. The second technique is by differentiation with respect to a parameter.