Integration by Parts

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Theorem

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.

Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\closedint a b$.


Then:

Primitive

$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$

on $\closedint a b$.


Definite Integral

$\ds \int_a^b \map f t \map G t \rd t = \bigintlimits {\map F t \map G t} a b - \int_a^b \map F t \map g t \rd t$


This technique is called integration by parts.


Also presented as

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 $u, v$ are functions of the independent variable.


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