  We discuss a non--commutative integration calculus arising in the
mathematical description of anomalies in fermion--Yang--Mills systems. We
consider the differential complex of forms
$u_0\ccr{\eps}{u_1}\cdots\ccr{\eps}{u_n}$ with $\eps$ a grading operator on a
Hilbert space $\cH$ and $u_i$ bounded operators on $\cH$ which naturally
contains the compactly supported de Rham forms on $\R^d$ (i.e.\ $\eps$ is the
sign of the free Dirac operator on $\R^d$ and $\cH$ a $L^2$--space on $\R^d$).
We present an elementary proof that the integral of $d$--forms
$\int_{\R^d}\trac{X_0\dd X_1\cdots \dd X_d}$ for $X_i\in\Map(\R^d;\gl_N)$, is
equal, up to a constant, to the conditional Hilbert space trace of $\Gamma
X_0\ccr{\eps}{X_1}\cdots\ccr{\eps}{X_d}$ where $\Gamma=1$ for $d$ odd and
$\Gamma=\gamma_{d+1}$ (`$\gamma_5$--matrix') a spin matrix anticommuting with
$\eps$ for $d$ even. This result provides a natural generalization of
integration of de Rham forms to the setting of Connes' non--commutative
geometry which involves the ordinary Hilbert space trace rather than the
Dixmier trace.
