Approximate solutions for the quantum bound states of an electron in graphene under a uniaxial strain modulation have been obtained. A first-order Taylor expansion for the Fermi velocity and the pseudo-vector potential in the effective Hamiltonian describing graphene under a uniaxial strain allows us to solve the corresponding differential equation of the eigenvalue problem. A finite number of bound states have been found and its spectrum is composed by non-standard Landau levels, that in contrast to the standard Landau levels, have a dependency on the square root of a second-order polynomial of the principal quantum number. A special case of interest is the periodic strain, where the zeroth-order approximation, that generates standard Landau levels, was derived. It turns out that the first-order approximation generates less energy levels than the zeroth-order approximation.