${F}_{1}(\ensuremath{\alpha})$ is defined as the contribution of the one-fermion-loop diagrams to the divergent part of the photon propagator in massless quantum electrodynamics. To sixth order, the perturbation expansion of ${F}_{1}(\ensuremath{\alpha})$ has rational coefficients: ${F}_{1}(\ensuremath{\alpha})=(\frac{2}{3})(\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}})+{(\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}})}^{2}\ensuremath{-}(\frac{1}{4}){(\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}}^{3}+\ensuremath{\cdots}$. It is not known whether the next term in this series is a rational number; however, we propose a new method, which uses integration by parts, for evaluating Feynman integrals which give rational numbers. Using this method we easily rederive the first three terms in the series for ${F}_{1}(\ensuremath{\alpha})$ and three other two-loop integrals, including the fourth-order correction to the vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\mu}}(p,p)$. We believe that our new integration techniques are powerful enough to evaluate the fourth term in the series for ${F}_{1}(\ensuremath{\alpha})$ if it is a rational number.