Primitive of Constant Multiple of Function

Theorem

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

Let $c$ be a constant.

Then:

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

Proof 1

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

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

$\blacksquare$

Proof 2

$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$

The result follows from the definition of primitive.