Reconstruction of unknown objects by microwave illumination requires efficient inversion for measured electromagnetic scattering data. In the integral equation approach for reconstructing dielectric objects based on the Born iterative method or its variations, the volume integral equations are involved because the imaging domain is fully inhomogeneous. When solving the forward scattering integral equation, the Nyström method is used because the traditional method of moments may be inconvenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and low requirement on geometrical discretization. When solving the inverse scattering integral equation, the Gauss-Newton minimization approach with a line search method (LSM) and multiplicative regularization method (MRM) is employed. The LSM can optimize the search of step size in each iteration, whereas the MRM may reduce the number of numerical experiments for choosing the regularization parameter. Numerical examples for reconstructing typical dielectric objects under limited observation angles are presented to illustrate the inversion approach.