Integral of Constant

Definite Integral
Let $$c$$ be a constant.

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

Proof for Definite Integral
Let $$f_c: \mathbb{R} \to \mathbb{R}$$ be the constant function.

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

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.