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In this book, we study some special cases of orthogonal polynomials and their associated orthogonal polynomials. They are beneficial in studying problems connected to solve extremal problem and to describe the approximation of continuous functions by these known orthogonal polynomials. In help we study finite linear combinations and finites summations sequences and calculating general summations related to special cases of orthogonal polynomials. These orthogonal polynomials are fruitful in applied to find the minimum value and the minimizing function for various definite integrals and solving extremal problems where their solutions are finites linear combinations of orthogonal polynomials. These orthogonal polynomials can be used to solve differential equations, whether they are linear or non-linear and to acquire numerical answers to differential equations, whether linear or nonlinear. The connections between the integral orthogonal polynomials, and Jacobi polynomials are highly helpful in obtaining the identities related to them. The integral orthogonal polynomials and their kernel polynomials are fruitful in approximation theory.