Integration by Parts/Primitive/Proof 2
Theorem
Integration by Parts is often seen presented in this sort of form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
or:
- $\ds \int u \rd v = u v - \int v \rd u$
where it is understood that $u$ and $v$ are functions of the independent variable.
Proof
Let $\map u x$ and $\map v x$ be integrable functions defined on $\closedint a b$.
Then:
\(\ds \map {\dfrac \d {\d x} } {u v}\) | \(=\) | \(\ds u \dfrac {\d v} {\d x} + v \dfrac {\d u} {\d x}\) | Product Rule for Derivatives | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds v \dfrac {\d u} {\d x}\) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {u v} - u \dfrac {\d v} {\d x}\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int v \dfrac {\d u} {\d x} \rd x\) | \(=\) | \(\ds \int \paren {\map {\dfrac \d {\d x} } {u v} - u \dfrac {\d v} {\d x} } \rd x\) | integrating both sides with respect to $x$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \int \map {\dfrac \d {\d x} } {u v} \rd x - \int u \dfrac {\d v} {\d x} \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds u v - \int u \dfrac {\d v} {\d x} \rd x\) | Fundamental Theorem of Calculus |
$\blacksquare$
Proof Technique
The technique of solving an integral in the form $\ds \int \map f t \map G t \rd t$ in this manner is called integration by parts.
Its validity as a solution technique stems from the fact that it may be possible to choose $f$ and $G$ such that $G$ is easier to differentiate than to integrate.
Thus the plan is to reduce the integral to one such that $\ds \int \map F t \map g t \rd t$ is easier to solve than $\ds \int \map f t \map G t \rd t$.
It may be, of course, that one or more further applications of this technique are needed before the solution can be extracted.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Integration by Parts
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integration by parts
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integration by parts