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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 ∈ (ℝ) is a test function, then fψ is distributionally integrable, and the formula $⟨,\psi ⟩=\left(\right){\int }_{-\infty }^{\infty }f\left(x\right)\psi \left(x\right)dx$, defines a distribution ∈ ’(ℝ)...