  In this paper we provide an alternative method to construct the Fueter
primitive of an axial monogenic function of degree $k$, which is complementary
to the one used in [F. Colombo, I. Sabadini and F. Sommen, The inverse Fueter
mapping theorem in integral form using spherical monogenics, Israel Journal of
Mathematics, 2012]. As a byproduct, we obtain an explicit description of the
kernel of the Fueter mapping. We also apply our method to obtain the Fueter
primitives of the Cauchy kernels with singularities on the unit sphere.
