The authors consider the inverse problem of determining the isotropic conductivity \( \sigma \) in \( \Omega \) from applications of an electric current \( j = \sigma \frac{\partial \Phi}{\partial n} \) on the boundary \( \partial \Omega \). Here, \( \Omega \subset \mathbb{R}^n, \, n = 2 \) or \( n = 3 \), denotes a bounded, simply connected domain, and \( \Phi \) denotes the electric potential. This problem is modelled by the equation \( \nabla \cdot (\sigma(x) \nabla \Phi(x)) = 0, x \in \Omega \), and is reformulated in terms of a pair of coupled integral equations, one of them being the first kind integral equation \( (AX)(x) = \int_{\Omega} \mathcal{H}(x,y) X(y) dy = \zeta(x) \). Here, \( \mathcal{H} \) is a bounded solution of the Schrödinger equation with respect to the second variable, i.e., \( \Delta_y \mathcal{H}(x,y) + \lambda(y) \mathcal{H}(x,y) = 0, x,y \in \Omega \), and \( \zeta \) can be represented by a boundary integral. The first kind integral equation \( AX = \zeta \) is approximately solved by mollifier methods, where the kernel \( \mathcal{H} \) is chosen appropriately to ensure existence of the mollifier. Further analysis is provided for \( \lambda \) fixed and the unit disk. Finally, results of some numerical experiments are presented.