In this paper, under certain conditions, we prove the existence of a unique solution of mixed integral equation of the second kind (MIESK) with singular kernel in the space    using Banach fixed point theorem. Then, using two numerical methods, we have a linear system of Fredholm integral equation (LSFIE). Moreover, we use Toeplitz matrix method and product Nystrom method to obtain a linear algebraic system (LAS) which can be solved numerically. The error estimate is discussed. Some examples are solved numerically.