Primitive of Function of Constant Multiple

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Theorem

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

Let $c$ be a constant.

Then:

$\displaystyle \int f \left({c x}\right) \ \mathrm d x = \frac 1 c \int f \left({u}\right) \ \mathrm d u$

where $u = c x$.


Proof

Let $u = c x$.

By Derivative of Identity Function: Corollary:

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


Thus:

\(\displaystyle \int f \left({c x}\right) \ \mathrm d x\) \(=\) \(\displaystyle \int \frac {f \left({u}\right)} c \ \mathrm d u\) Primitive of Composite Function‎
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 c \int f \left({u}\right) \ \mathrm d u\) Primitive of Constant Multiple of Function

$\blacksquare$


Sources