In this paper general class of Beta integrals is considered for particular function . This class is further used to evaluate certain integrals for some special functions. The class of integrals studied in this paper is integral involving the product of several exponential functions and Gauss’s hypergeometric function. As the application of this general integral some integrals for some particular functions are derived.

We have studied general class of Beta integrals for particular functions in. This class is used to evaluate certain integrals for some special functions by assigning the different- different forms of the functions. Various known and new simpler integral formulae are obtained from our main results. As the application of these simpler results many interesting consequences are derived. The methodology and technique that we have used in this paper to obtain the results can be further extended for more general new functions. The young researchers can use our finding to develop new study.

The Beta function is a unique function where it is classified as the first kind of Euler’s integral. The Beta function is defined in the domains of real numbers. The notation to represent the beta function is “β”. The beta function is meant by B(p, q), where the parameters p and q should be real numbersThe Beta function was introducedearly 19th century.The function was later extensively studied by other mathematicians, including Augustin-Louis Cauchy and CarlGustav Jacobi, who contributed to understanding its propertiesand  relationship with other mathematical functions.

The Gamma function was first introduced by Swiss mathematicianLeonhard Euler in the 18th centuryalthough it was later formalized and studied extensively by other mathematicians

Both functions have since been further developed and studied by numerous mathematicians and have found applications in variousareas ofMathematics, Physics and engineering.Due to usefulness of these functions Chaudhary and Zubair studiedextended Gamma and Beta functions

The Pochhammer symbol, also known as the rising factorial, was named after the German mathematician Leo August Pochhammer. Pochhammer lived from 1841 to 1920 and made significant contributions to various areas of mathematics, including number theory, algebra, and analysis.

The Gauss hypergeometric function ₂F₁ is a special function that plays a fundamental role in many areas of mathematics, physics, and engineering. It is named after the German mathematician Carl Friedrich Gauss.The Gauss hypergeometric function has numerous properties and relationships with other mathematical functions, and its study continues to be an active area of research in mathematics and its applications.

F₁, F₂, F₃, and F₄ are the functions that Appell developed in 1880, now popularly known as Appell functions. These are two variables extension of ₂F₁  function.

It is useful here to define the following relations for Pocchammer symbol[10], pp.21-23; eqs. (4), (15),

Main Result

In this section we will study the following general class of integrals in view of the series integral representation of extended Beta function in (1.7)

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