Integral of Constant

Theorem
Let $$c$$ be a constant.

Definite Integral
$$\int_a^b c \, dx = c \left({b-a}\right)$$.

Indefinite Integral
$$\int c \, dx = c x + C$$ 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 $$c \left({b-a}\right) \le \int_a^b c \, dx \le c \left({b-a}\right)$$.

Hence the result.

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

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 a Constant, $$D_x \left({c x + C}\right) = c$$.

Hence the result.