Primitive of Constant Multiple of Function/Proof 1

Theorem

Let $f$ be a real function which is integrable.

Let $c$ be a constant.

Then:

$\ds \int c \map f x \rd x = c \int \map f x \rd x$

Proof

From Linear Combination of Primitives:

$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$

The result follows by setting $\lambda = c$ and $\mu = 0$.

$\blacksquare$