Significance of function $I$ such that $\omega = I(d\omega) + d(I\omega)$ for differential forms

Question

In Calculus on Manifolds, Spivak proves the Poincaré Lemma by introducing a function $I$ which maps $p$-forms to $(p-1)$-forms (for each $p$). Suppose there is a differential form $\omega\in \Omega^p (\mathbb{R}^n)$ defined on a star-shaped region in $\mathbb{R}^n$ centered around the origin. If $\omega$ is given by $$\omega=\sum_{i_1<\cdots<i_p} \omega_{i_1,\ldots,i_p}dx^{i_1}\wedge \cdots \wedge dx^{i_p},$$ Spivak defines $$I\omega(x)=\sum_{i_1<\cdots<i_p}\sum_{\alpha=1}^p (-1)^{\alpha -1}\left(\int_0^1 t^{p-1}\omega_{i_1,\ldots,i_p}(tx)\,dt\right)x^{i_\alpha}\,dx^{i_1}\wedge \cdots \wedge \widehat{dx^{i_\alpha}}\wedge\cdots\wedge dx^{i_p}.$$ (In the expression above, the symbol $\widehat\quad$ over $dx^{i_\alpha}$ indicates that it is omitted.) Spivak goes on to demonstrate that $\omega = I(d\omega) + d(I\omega)$. Although I follow the proof, the definition of $I$ seems to come out of nowhere. Does $I$ have any deeper significance? It is never mentioned again anywhere else in the book.