A discretization method is proposed where a tunable integration scheme is applied to the finite element method (FEM). The method is characterized by one continuous parameter, p. A theoretical error analysis is given and three different eigenvalue problems are used as test cases: a simple example with constant coefficients and two model problems from ideal and resistive magnetohydrodynamics. It is shown that, for judicious choices of p, the tunable integration method clearly improves the convergence of the strict FEM. The sensitivity to the choice of integration parameter is discussed.