  We show that direct Feynman-parametric loop integration is possible for a
large class of planar multi-loop integrals. Much of this follows from the
existence of manifestly dual-conformal Feynman-parametric representations of
planar loop integrals, and the fact that many of the algebraic roots associated
with (e.g. Landau) leading singularities are automatically rationalized in
momentum-twistor space---facilitating direct integration via partial
fractioning. We describe how momentum twistors may be chosen non-redundantly to
parameterize particular integrals, and how strategic choices of coordinates can
be used to expose kinematic limits of interest. We illustrate the power of
these ideas with many concrete cases studied through four loops and involving
as many as eight particles. Detailed examples are included as ancillary files
to this work's submission to the arXiv.
