Primitive of Composite Function
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Theorem
Let $f$ and $g$ be a real functions which are integrable.
Let the composite function $g \circ f$ also be integrable.
Then:
\(\ds \int \map {\paren {g \circ f} } x \rd x\) | \(=\) | \(\ds \int \map g u \frac {\d x} {\d u} \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\map g u} {\map {f'} x} \rd u\) |
where $u = \map f x$.
Corollary
Let $f$ and $g$ be a real functions which are integrable.
Let the composite function $g \circ f$ also be integrable.
Then:
\(\ds \int \map {\paren {g \circ f} } x \map {f'} x \rd x\) | \(=\) | \(\ds \int \map g u \rd u\) |
where $u = \map f x$.
Proof
\(\ds \map F x\) | \(=\) | \(\ds \int \map {\paren {g \circ f} } x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \map g {\map f x} \rd x\) | Definition of Composition of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \map g u \rd x\) | where $u = \map f x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d F} {\d x}\) | \(=\) | \(\ds \map g u\) | Definition of Primitive (Calculus) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d F} {\d u} \frac {\d u} {\d x}\) | \(=\) | \(\ds \map g u\) | Chain Rule for Derivatives | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d F} {\d u} \frac \d {\d x} \map f x\) | \(=\) | \(\ds \map g u\) | Definition of $u$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d F} {\d u} \map {f'} x\) | \(=\) | \(\ds \map g u\) | Definition of Derivative | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d F} {\d u}\) | \(=\) | \(\ds \frac {\map g u} {\map {f'} x}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds F\) | \(=\) | \(\ds \int \frac {\map g u} {\map {f'} x} \rd u\) | Definition of Primitive (Calculus) | ||||||||||
\(\ds \) | \(=\) | \(\ds \int \map g u \frac {\d x} {\d u} \rd u\) | Derivative of Inverse Function: $\dfrac {\d x} {\d u} = \dfrac 1 {\d u / \d x}$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.6$