Context. The smoothed particle hydrodynamics (SPH) technique is a well-known numerical method that has been applied to simulating the evolution of a wide variety of systems. Modern astrophysical applications of the method rely on the Lagrangian formulation of fluid Euler equations, which is fully conservative. A different scheme, based on a matrix approach to the SPH equations is currently being used in computational fluid dynamics. An original matrix formulation of SPH based on an integral approach to the derivatives, called IAD0, has been recently proposed and is fully conservative and well-suited to simulating astrophysical processes. Aims. The behavior of the IAD0 scheme is analyzed in connection with several astrophysical scenarios, and compared to the same simulations carried out with the standard SPH technique. Methods. The proposed hydrodynamic scheme is validated using a variety of numerical tests that cover important topics in astrophysics, such as the evolution of supernova remnants, the stability of self-gravitating bodies, and the coalescence of compact objects. Results. The analysis of the hydrodynamical simulations of the above-mentioned astrophysical scenarios suggests that the SPH scheme built with the integral approach to the derivatives improves the results of the standard SPH technique. In particular, there is a better development of hydrodynamic instabilities, a good description of self-gravitating structures in equilibrium and a reasonable description of the process of coalescence of two white dwarfs. We also observed good conservations of energy and both linear and angular momenta that were generally better than those of standard SPH. In addition the new scheme is less susceptible to pairing instability. Conclusions. We present a formalism based on a tensor approach to Euler SPH equations that we checked using a variety of three-dimensional tests of astrophysical interest. This new scheme is more accurate because of the re-normalization imposed on the interpolations, which is fully conservative and less prone to undergoing the pairing instability. The analysis of these test cases suggests that the method may improve the simulation of many astrophysical problems with only a moderate computational overload.