###### Abstract  
This article is concerned with the construction of approximate analytic solutions to linear Fredholm integral equations of the second kind with general continuous kernels. A unified treatment of some classes of analytical and numerical classical methods, such as the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM) and the Projection Methods (PM), is proposed. The problem is formulated as an abstract equation in a Banach space and a solution formula is derived. Then, several approximating schemes are discussed. In all cases, the method yields an explicit, albeit approximate, solution. Several examples are solved to illustrate the performance of the technique.  
Fredholm integral equations; direct method; numerical methods; symbolic computations; degenerate kernels; non-separable kernels; power series; interpolation; quadrature 4.0.  
Methods (DKM), the Quadrature Methods (QM), and the Projection Methods (PM); see, for example, the standard treatises [5; 6; 7], the traditional articles [10; 11; 12], and the recent papers [13; 14; 15; 16; 17; 18; 19; 20; 21]. The DCM has the advantage that it delivers the exact solution in closed form, but its application is limited to special cases where the kernel is separable (degenerate) and the integrals involved can be determined analytically. DKM utilize approximate finite representations for the kernel, and possibly the input function, and they are easy to manage and to perform error analysis. However, as with DCM, when the terms in the degenerate kernel are other than simple functions, then their integration has to be performed numerically. QM are very efficient, particularly when they are combined with [PERSON]'s interpolation, although their error analysis becomes more involved. The formulation of PM is more complicated, while some of these methods can be dealt with as the DKM. The above methods are treated separately in the literature.  
Here, we present a common formulation suitable for symbolic computations for all of these techniques. Our approach is based on the idea that in all instances, the integral equation in (1) may be cast or reduced to an equation of the form  
\[u(x)-\lambda\sum_{j=1}^{m}g_{j}(x)\Psi_{j}(u)=f(x),\quad x\in[a,b], \tag{2}\]  
where for \(m\geq 1\), \(g_{j}(x)\), \(j=1,2,\ldots,m\), are known continuous functions on \([a,b]\) and \(\Psi_{j}\), \(j=1,2,\ldots,m\) are linear bounded functionals such as definite integrals or sums of values at some points contained in \([a,b]\). Both \(\{g_{j}(x)\}\) and \(\{\Psi_{j}\}\) are obtained through the separation or approximation of the kernel, the approximation of the integral, the unknown function or a combination of them. Equation (2), under certain conditions, which are associated with the existence and uniqueness of the solution of the integral equation in (1), can be solved symbolically to obtain an exact or approximate analytic solution of (1).  
We implement the proposed method to construct exact closed-form solutions when the kernel \(K(x,t)\) is separable, approximate analytic solutions when \(K(x,t)\) is not separable, but it can be represented as a truncated power series or an interpolation polynomial, and semi-discrete solutions when the definite integral is replaced by a finite sum by using a quadrature rule. The economy and the efficiency of the method are revealed by solving several tests problems from the literature.  
The paper is organized as follows. In Section 2, an abstract formulation of the problem in a Banach space is presented and a closed-form solution of (2) is derived. In Sections 3-6, we elaborate on the cases where \(K(x,t)\) is separable, \(K(x,t)\) is approximated by a power series or a polynomial, the definite integral is replaced by a quadrature formula and the unknown function is approximated by a polynomial, respectively. Several examples are solved in Section 7. Finally, some conclusions are quoted in Section 8.