Euler's Integral, often referred to in the context of the Gamma function, is a mathematical expression that extends the concept of factorials to non-integer values. It is defined as the integral from 0 to infinity of t^x-1 e^-t \, dt , where x is a complex number. This integral converges for positive values of x and provides a way to calculate factorials for non-integer numbers, such as \Gamma(n) = (n-1)! for positive integers n . The significance of Euler's Integral lies in its applications across various fields, including probability theory, statistics, and complex analysis. It allows for the computation of probabilities and distributions that involve non-integer parameters. Additionally, it plays a crucial role in solving differential equations and in the study of special functions, making it a fundamental tool in advanced mathematics.