Integral of Constant

Theorem
Let $c$ be a constant.

Definite Integral

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

Indefinite Integral

 * $\displaystyle \int c \ \mathrm dx = c x + C$ where $C$ is an arbitrary constant.

Proof for Definite Integral
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.

Proof for Indefinite Integral
Let $\displaystyle F \left({x}\right) = \int c \ \mathrm dx$

Then from the definition of indefinite integral, $F' \left({x}\right) = c$.

From Derivative of Function of Constant Multiple, $D_x \left({c x}\right) = c$.

From Primitives which Differ by Constant, $D_x \left({c x + C}\right) = c$.

Hence the result.