Integration by Parts

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Theorem

Let $f$ and $g$ be real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\left[{a \,.\,.\, b}\right]$.


Then:

$\displaystyle \int_a^b f \left({t}\right) G \left({t}\right) \rd t = \left[{F \left({t}\right) G \left({t}\right)}\right]_a^b - \int_a^b F \left({t}\right) g \left({t}\right) \rd t$


This is frequently written as:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \ \frac {\d u} {\d x} \rd x$

or:

$\displaystyle \int u \rd v = u v - \int v \rd u$

where it is understood that $u, v$ are functions of the independent variable.


Proof

By Product Rule for Derivatives:

$D \left({FG}\right) = f G + F g$

Thus $F G$ is a primitive of $f G + F g$ on $\left[{a \,.\,.\, b}\right]$.

Hence, by the Fundamental Theorem of Calculus:

$\displaystyle \int_a^b \left({f \left({t}\right) G \left({t}\right) + F \left({t}\right) g \left({t}\right)}\right) \rd t = \left[{F \left({t}\right) G \left({t}\right)}\right]_a^b$

The result follows.

$\blacksquare$


Notes

The technique of solving an integral in the form $\displaystyle \int_a^b f \left({t}\right) G \left({t}\right) \rd t$ 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 integration to one such that $\displaystyle \int_a^b F \left({t}\right) g \left({t}\right) \rd t$ is easier to solve than $\displaystyle \int_a^b f \left({t}\right) G \left({t}\right) \rd t$

It may be, of course, that a further application of this technique is needed before the solution can be extracted.


Sources