The formal derivative of a polynomial $p \in R[T]$ over a commutative ring $R$ has a conceptual definition using the ring of dual numbers $R[T][\varepsilon]$: $$p(T + \varepsilon) = p(T) + p'(T) \varepsilon$$ Is there also a conceptual definition of the formal integral$$\int_0^T p(X) \, dX ~ ?$$ Here $R$ is a commutative $\mathbb{Q}$-algebra. Any definition which is more conceptual than the ad hoc definition $\int_0^T X^n \, dX = \frac{1}{n+1} T^{n+1}$ is appreciated. Ideally, the first fundamental theorem of calculus $$\int_0^{T+\varepsilon} p(X) \, dX = \int_0^T p(X) \, dX + p(T) \varepsilon$$ should follow directly from the definition, including $\int_0^0 p(X) \, dX = 0$.
In synthetic differential geometry and smooth infinitesimal analysis, a similar definition of the derivative works for arbitrary functions $\mathbb{R} \to \mathbb{R}$. But for integrals, there is just an axiom that they exist and make the first fundamental theorem of calculus true (Bell, A primer of infinitesimal analysis, Section 6.1). I would prefer a formula which is similar to the implicit defintion of the derivative, at least for polynomials.