{
  "abstract": "The \"Riemann-type integrals\" of the title are not a single new integral, but the family of classical constructions -- Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô -- unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on $\\mathbb{R}$; it vanishes in the limit under a quadratic-variation condition $Q(π)\\to0$ on $\\mathbb{C}$ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction -- except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point -- that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts -- as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on $\\mathbb{R}$ and $\\mathbb{C}$; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05562",
  "author_total": 1,
  "authors": [
    {
      "name": "Luciano Tubaro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/az2yt4fka780c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05562",
  "fulltext_url": null,
  "id": "az2yt4fka780c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05562",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2610|Mathematical and Theoretical Analysis"
    ]
  ],
  "openalex": "W7212068116",
  "pdf_url": "https://arxiv.org/pdf/2609.05562",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/az2yt4fka780c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.05562",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05562"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Mathematical and Theoretical Analysis",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "A simple method of computing Riemann-type integrals",
  "title_text": "A simple method of computing Riemann-type integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/az2yt4fka780c",
  "venue": "arXiv",
  "versions": []
}