Integration on Polynomials is Linear Operator

Theorem
Let $$P \left({\R}\right)$$ be the vector space of all polynomial functions on the real number line $$\R$$.

Let $$S$$ be the mapping defined as:
 * $$\forall p \in P \left({\R}\right): \forall x \in \R: S \left({p \left({x}\right)}\right) = \int_0^x p \left({t}\right) \mathrm d t$$

Then $$S$$ is a linear operator on $$P \left({\R}\right)$$.

Proof
Proved in Linear Combination of Integrals.