Integral of Constant/Definite

Theorem
Let $c$ be a constant.
 * $\displaystyle \int_a^b c \ \ \mathrm dx = c \left({b-a}\right)$.

Proof
Let $f_c: \R \to \R$ be the constant function.

By definition:
 * $\forall x \in \R: f_c \left({x}\right) = c$

Thus:
 * $\sup \left({f_c}\right) = \inf \left({f_c}\right) = c$

So from Upper and Lower Bounds of Integral‎, we have:
 * $\displaystyle c \left({b-a}\right) \le \int_a^b c \ \ \mathrm dx \le c \left({b-a}\right)$

Hence the result.