  We show that every semialgebraic set admits a semialgebraic triangulation
such that each closed simplex is $C^1$ differentiable. As an application, we
give a straightforward definition of the integration $\int_X \omega$ over a
compact semialgebraic subset $X$ of a differential form $\omega$ on an ambient
algebraic manifold, that provides a significant simplification of the theory of
semialgebraic singular chains and integrations. Our results hold over every
(possibly non-archimedian) real closed field.
