# Integration by Parts/Definite Integral

## 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:

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

## Proof

By Product Rule for Derivatives:

- $\map D {F G} = f G + F g$

Thus $F G$ is a primitive of $f G + F g$ on $\closedint a b$.

Hence, by the Fundamental Theorem of Calculus:

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

The result follows.

$\blacksquare$

## 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

- 1967: Michael Spivak:
*Calculus*... (previous) ... (next): Part $\text {III}$: Derivatives and Integrals: Chapter $18$: Integration in Elementary Terms: Theorem $1$ (Integration by Parts) - 1976: K. Weltner and W.J. Weber:
*Mathematics for Engineers and Scientists*... (previous) ... (next): $6$. Integral Calculus: Appendix: Rules and Techniques of Integration: $2.1$*Integration by parts* - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.21$