# Primitive of Function of Constant Multiple

## Theorem

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

Let $c$ be a constant.

Then:

$\displaystyle \int \map f {c x} \rd x = \frac 1 c \int \map f u \d u$

where $u = c x$.

## Proof

Let $u = c x$.

$\dfrac {\d u} {\d x} = c$

Thus:

 $\ds \int \map f {c x} \rd x$ $=$ $\ds \int \frac {\map f u} c \rd u$ Primitive of Composite Function‎ $\ds$ $=$ $\ds \frac 1 c \int \map f u \rd u$ Primitive of Constant Multiple of Function

$\blacksquare$