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: $14.5$