  Scattering amplitudes in Yang-Mills theory can be represented in the
formalism of Cachazo, He and Yuan (CHY) as integrals over an auxiliary
projective space---fully localized on the support of the scattering equations.
Because solving the scattering equations is difficult and summing over the
solutions algebraically complex, a method of directly integrating the terms
that appear in this representation has long been sought. We solve this
important open problem by first rewriting the terms in a manifestly
Mobius-invariant form and then using monodromy relations (inspired by analogy
to string theory) to decompose terms into those for which combinatorial rules
of integration are known. The result is a systematic procedure to obtain
analytic, covariant forms of Yang-Mills tree-amplitudes for any number of
external legs and in any number of dimensions. As examples, we provide compact
analytic expressions for amplitudes involving up to six gluons of arbitrary
helicities.
