Primitive of Function of Constant Multiple

Theorem

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

Let $c$ be a constant.

Then:

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

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

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.5$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.5.$
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