Primitive of Function of Constant Multiple

From ProofWiki
Jump to navigation Jump to search

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 $u = c x$.

By Derivative of Identity Function: Corollary:

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


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

$\blacksquare$


Sources