{
  "abstract": "In this note, we address a traditional pedagogical asymmetry between algorithmic differentiation and heuristic integration by introducing an extra parametric-order differential framework. By treating the order of differentiation as a parametric axis (discrete or continuous), classical integration is reframed as a local differential evaluation at the non-positive boundary level. This analytical shift alleviates some traditional routines of integration by parts, by replacing the quadrature of a product of elementary functions with a direct differential series that either truncates into a finite algebraic sum or resolves through cyclic symmetries. This parametric-order approach also bridges the gap with fractional calculus, showing that non-integer derivatives emerge as a seamless extension of the same operational dimension. By reducing global quadrature to local differentiation, this note offers an efficient, conceptual tool for undergraduate analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40681",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexandre M. Afonso"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4sknfctdtjw4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40681",
  "fulltext_url": null,
  "id": "b4sknfctdtjw4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40681",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40681",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4sknfctdtjw4.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40681",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40681"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Integration by differentiation",
  "title_text": "Integration by differentiation",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/b4sknfctdtjw4",
  "venue": "arXiv",
  "versions": []
}