Integration by differentiation
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.
Related
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- Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
- Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots
- Smooth periodic extensions by extrapolation and blending for FFT-based differentiation
- Numerical differentiation of functions on the half-line in a weighted uniform metric
BibTeX
@misc{afonso2026integration,
title = {{Integration by differentiation}},
author = {Alexandre M. Afonso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40681},
eprint = {2610.40681},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40681},
}
RIS
TY - UNPB
TI - Integration by differentiation
AU - Alexandre M. Afonso
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.40681
UR - https://arxiv.org/abs/2610.40681
AB - 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.
ER -