  We calculate the integrated trace anomaly for a real spin-0 scalar field in
six dimensions in a torsionless curved space without a boundary. We use a path
integral approach for a corresponding supersymmetric quantum mechanical model.
Weyl ordering the corresponding Hamiltonian in phase space, an extra two-loop
counterterm ${1/8}\bigg(R + g^{ij} \Gamma^{l}_{k i} \Gamma^{k}_{l j} \bigg)$ is
produced in the action. Applying a recursive method we evaluate the components
of the metric tensor in Riemann normal coordinates in six dimensions and
construct the interaction Langrangian density by employing the background field
method.
  The calculation of the anomaly is based on the end-point scalar propagator
and not on the string inspired center-of-mass propagator which gives incorrect
results for the local trace anomaly. The manipulation of the Feynman diagrams
is partly relied on the factorization of four dimensional subdiagrams and
partly on a brute force computer algebra program developed to serve this
specific purpose. The computer program enables one to perform index
contractions of twelve quantum fields (10395 in the present case) a task which
cannot be accomplished otherwise. We observe that the contribution of the
disconnected diagrams is no longer proportional to the two dimensional trace
anomaly (which vanishes in four dimensions). The integrated trace anomaly is
finally expressed in terms of the 17 linearly independent scalar monomials
constructed out of covariant derivatives and Riemann tensors.
