Operational integration of ∫ P(x) cos(ωx) dx and ∫ P(x) sin(ωx) dx: complexification, nilpotency, and finite algebraic reduction

An operational treatment of the integrals ∫ P(x) sin(ωx) dx and ∫ P(x) cos(ωx) dx is presented, where P ∈ Vₙ is a polynomial of degree at most n and ω ≠ 0.

The method proceeds by complexification: replacing cos(ωx) and sin(ωx) with the real and imaginary parts of eⁱʷˣ, the problem reduces to inverting the operator D + iω on Vₙ.

Since D is nilpotent of order n + 1 on Vₙ, the Neumann series associated with (D + iω)⁻¹ truncates exactly and yields closed antiderivatives without recursion.

The contribution of this work is structural rather than computational. It shows that a classical family of trigonometric integrals, normally treated by successive integrations by parts, can be understood as a case of finite operational inversion on a nilpotent algebra, with the complex exponential acting as a diagonalizing element that unifies both families.

This reinterpretation is based on the general operational framework in which the integration-by-parts formula is obtained as a consequence of the Leibniz rule through the identity JD = I − Π, developed in Moya (2026, DOI: 10.5281/zenodo.20513922). All results have been formally verified in Lean 4 (Zenodo, DOI: 10.5281/zenodo.20563852).