# Integration by Parts

## 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

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

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 $\displaystyle \int \map f t \map G t \rd t$ 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 $\displaystyle \int \map F t \map g t \rd t$ is easier to solve than $\displaystyle \int \map f t \map G t \rd t$.

It may be, of course, that one or more further applications of this technique are needed before the solution can be extracted.