$$ \phi(x) = \sum_{i=1}^{n+1} c_{i-1} B_{i-1,n}(x), \tag{1.3} where $c_{i-1}$ ($i=1,2,\dots,n+1$)'s are unknown constants and $B_{i-1,n}(x)$ ($i=1,2,\dots,n+1$) are Bernstein polynomials of degree $n$ defined on an interval $(\alpha, \beta)$, and are given by B_{i-1,n}(x) = \binom{n}{i-1} \frac{(x-\alpha)^{i-1}(\beta-x)^{n-i+1}}{(\beta-\alpha)^n}, \quad i=1,2,\dots,n+1 \tag{1.4} \sum_{i=1}^{n+1} c_{i-1} \Psi_{i-1}(x) = f(x), \quad \alpha < x < \beta, \qquad (1.5) \Psi_{i-1}(x) = B_{i-1,n}(x) + \int_{\alpha}^{\beta} k(x,t) B_{i-1,n}(t) dt. \quad (1.6) The determination of the solution (1.3) can then be completed by solving the over-determined system of linear algebraic equations (1.5) for the unknown constants $c_{i-1}$ ($i = 1, 2, \dots, n+1$). The best solution of the over-determined system of Eq. (1.5) is obtainable by the method of least-squares giving rise to the system of determinate linear algebraic equations, as given by: \sum_{i=1}^{n+1} c_{i-1} D_{ij} = B_j, \quad j = 1, 2, \dots, n+1, \tag{1.7} D_{ij} = \int_{\alpha}^{\beta} \Psi_{i-1}(x) \Psi_{j-1}(x) dx, \qquad (1.8) B_j = \int_\alpha^\beta f(x) \Psi_{j-1}(x) dx. \tag{1.9} The above system of Eq. (1.7) is different from the one obtained by Mandal and Bhattacharya [1], which was derived by multiplying the relation (1.5) by $B_{j-1,n}(x)$ and integrating. We observe that the above procedure of the determination of the coefficients $c_{i-1}$ ($i = 1, 2, \dots, n+1$) gives rise to computational difficulties because of the fact that a large number of integrals need to be evaluated which involve the Bernstein polynomials, even by selecting $n$ to be as small as $n = 4$. We have avoided these difficulties by recasting the expression (1.3) as \phi(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1} x^{n-1} + a_n x^n, \tag{1.10} where, if $\alpha = 0, \beta = 1$, we get a_0 = c_0, a_1 = -nc_0 + nc_1, a_2 = \frac{n(n-1)}{2} {c_0 + c_2} - n(n-1)c_1, a_{n-1} = (-1)^{n-1}nc_0 + (-1)^{n-2}n(n-1)c_1 + (-1)^{n-3}\frac{n(n-1)(n-2)}{2}c_2 + \cdots + nc_{n-1}, a_n = (-1)^n c_0 + (-1)^{n-1} n c_1 + (-1)^{n-2} \frac{n(n-1)}{2} c_2 + \cdots - n c_{n-1} + c_n. \phi(x) = \sum_{i=1}^{N+1} a_{i-1} x^{i-1}, \tag{1.11} $$
where $a_{i-1}$ ($i = 1, \dots, N+1$) are unknown constants to be determined then it amounts to determining the values of $\phi(x)$ at $N+1$ points in its domain of definition (see interpolation formula). This forces us to approximate the integral term (see relations(1.1) and (1.2)) of the integral equation by a suitable quadrature formula requiring the knowledge of these $(N+1)$ values of $\phi$.