Integral operators are mathematical tools that transform functions by integrating them against a kernel function. They are often represented in the form (Kf)(x) = \int K(x, y) f(y) \, dy , where K(x, y) is the kernel and f(y) is the function being transformed. Integral operators are widely used in various fields, including functional analysis, quantum mechanics, and signal processing. These operators can be classified into different types, such as Volterra and Fredholm operators, based on their properties and the nature of the kernel. They play a crucial role in solving differential equations and in the study of Hilbert spaces, providing a framework for understanding complex mathematical problems through simpler integral forms.