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A General Integral

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 $ψ\in\mathcal{D}(\mathbb{R}%) $ is a test function, then $fψ$ is distributionally integrable, and the formula% [<\mathsf{f},ψ> =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) ψ(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, subst