Operational integration of ∫xⁿeᵃˣdx: nilpotency, finite Neumann series and algebraic reduction of integration by parts

This article presents a rigorous operational derivation of the integral ∫xⁿeᵃˣdx, n∈ℕ₀, a≠0, by studying the action of the differentiation operator D=d/dx on the finite-dimensional space Vₙ of polynomials of degree at most n. The central idea is to reinterpret the integration of polynomial-exponential functions as the inversion of the operator D+a on Vₙ. Since Dⁿ⁺¹=0 on Vₙ, the Neumann series associated with (D+a)⁻¹ is not an infinite expansion or an approximation, but an exact finite sum. This yields the closed formula ∫xⁿeᵃˣdx = eᵃˣ Σₖ₌₀ⁿ (−1)ᵏ n! xⁿ⁻ᵏ / [aᵏ⁺¹(n−k)!] + C. The formula is classically obtainable by repeated integration by parts, but here it is derived from a structural principle: the nilpotency of the differential operator on polynomial spaces. The contribution of the article is not the discovery of a new antiderivative, but the demonstration that a classical family of integrals can be solved through a finite algebraic identity, avoiding the recursive nature of integration by parts and providing a systematic operational formulation.