  In this work we obtain the planar free energy for the Hermitian one-matrix
model with various choices of the potential. We accomplish this by applying an
approach that bypasses the usual diagonalization of the matrices and the
introduction of the eigenvalue density, to directly zero in the evaluation of
the planar free energy. In the first part of the paper, we focus on potentials
with finitely many terms. For various choices of potentials, we manage to find
closed expressions for the planar free energy, and in some cases determine or
bound their radius of convergence as a series in the 't Hooft coupling. In the
second part of the paper we consider specific examples of potentials with
infinitely many terms, that arise in the study of ${\cal N}=2$ super Yang-Mills
theories on $S^4$, via supersymmetric localization. In particular, we manage to
write the planar free energy of two non-conformal examples: SU(N) with
$N_f<2N$, and ${\cal N}=2^*$.
