Primitive of Constant

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Theorem

Let $c$ be a constant.

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


Proof

Let:

$\displaystyle F \left({x}\right) = \int c \ \mathrm dx$

From the definition of primitive:

$F' \left({x}\right) = c$

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.

$\blacksquare$


Sources