Integration of a Constant
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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
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.
$\blacksquare$
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 a Constant, $D_x \left({c x + C}\right) = c$.
Hence the result.
$\blacksquare$
