  Calculating the long term solution of ordinary differential equations, such
as those of the $N$-body problem, is central to understanding a wide range of
dynamics in astrophysics, from galaxy formation to planetary chaos. Because
generally no analytic solution exists to these equations, researchers rely on
numerical methods which are prone to various errors. In an effort to mitigate
these errors, powerful symplectic integrators have been employed. But
symplectic integrators can be severely limited because they are not compatible
with adaptive stepping and thus they have difficulty accommodating changing
time and length scales. A promising alternative is time-reversible integration,
which can handle adaptive time stepping, but the errors due to time-reversible
integration in astrophysics are less understood. The goal of this work is to
study analytically and numerically the errors caused by time-reversible
integration, with and without adaptive stepping. We derive the modified
differential equations of these integrators to perform the error analysis. As
an example, we consider the trapezoidal rule, a reversible non-symplectic
integrator, and show it gives secular energy error increase for a pendulum
problem and for a H\'{e}non---Heiles orbit. We conclude that using reversible
integration does not guarantee good energy conservation and that, when
possible, use of symplectic integrators is favored. We also show that
time-symmetry and time-reversibility are properties that are distinct for an
integrator.
