The efficient computation of molecular integrals and their derivatives is a crucial step in molecular property evaluation in modern quantum chemistry. In this study, an application programming interface (API) framework, denoted Molint, for the computation of molecular integrals and their first derivatives, over contracted Gaussian functions, for the density-fitted methods is reported. The Molint is free software and it includes overlap, dipole, kinetic, potential, 2-index, and 3-index integrals, and their first derivatives. Furthermore, the Molint provides a smooth approach to build the Fock matrix and evaluate analytic gradients for the density-fitted methods. The Molint is a C++/Fortran hybrid code, which can take advantage of shared-memory parallel programming techniques. Our results demonstrate that the Molint is an efficient and user-friendly API for the computation of molecular integrals and their first derivatives.