  We define an integral, the distributional integral of functions of one real
variable, that is more general than the Lebesgue and the
Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of
functions with distributional values everywhere or nearly everywhere.
  Our integral has the property that if $f$ is locally distributionally
integrable over the real line and $\psi\in\mathcal{D}(\mathbb{R}%) $ is a test
function, then $f\psi$ is distributionally integrable, and the formula%
[<\mathsf{f},\psi> =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) \psi(x)
\,\mathrm{d}% x\,,] defines a distribution
$\mathsf{f}\in\mathcal{D}^{\prime}(\mathbb{R}) $ that has distributional point
values almost everywhere and actually $\mathsf{f}(x) =f(x) $ almost everywhere.
  The indefinite distributional integral $F(x) =(\mathfrak{dist})
\int_{a}^{x}f(t) \,\mathrm{d}t$ corresponds to a distribution with point values
everywhere and whose distributional derivative has point values almost
everywhere equal to $f(x).$
  The distributional integral is more general than the standard integrals, but
it still has many of the useful properties of those standard ones, including
integration by parts formulas, substitution formulas, even for infinite
intervals --in the Ces\`{a}ro sense--, mean value theorems, and convergence
theorems. The distributional integral satisfies a version of Hake's theorem.
Unlike general distributions, locally distributionally integrable functions can
be restricted to closed sets and can be multiplied by power functions with real
positive exponents.
