# Integration by Parts (Partial integration)

Integration by parts (or partial integration) is one of the most useful basic calculus operations. It is a generalization of the fundamental theorem of calculus for the case of the product of two functions.

## Fundamental theorem of Calculus

The classical definition of the definite integral of a function ${\displaystyle F(x)={\frac {df}{dx}}}$ on the closed interval ${\displaystyle [a,b]}$ as [1]

${\displaystyle \int _{a}^{b}F(x)\;dx=\int _{a}^{b}{\frac {df(x)}{dx}}\;dx=f(b)-f(a)}$

## Integration by parts

The second important result from calculus is the form of the Fundamental Theorem known as integration by parts or partial integration. If we consider two differentiable functions ${\displaystyle u(x),v(x)}$, then we may write by the product rule

${\displaystyle {\frac {d}{dx}}\left(u(x)v(x)\right)={\frac {du}{dx}}v(x)+u(x){\frac {dv}{dx}}}$

Integrating both sides from ${\displaystyle a}$ to ${\displaystyle b}$ yields

${\displaystyle \int _{a}^{b}\left[{\frac {d}{dx}}\left(u(x)v(x)\right)\right]\;dx=\int _{a}^{b}\left[{\frac {du}{dx}}v(x)+u(x){\frac {dv}{dx}}\right]\,dx}$

which may be written more simply as

${\displaystyle \int _{a}^{b}\left[{\frac {d}{dx}}\left(u(x)v(x)\right)\right]\;dx=\int _{a}^{b}v\;du+\int _{a}^{b}u\;dv.}$

Recognizing that the term on the left is an exact differential, we may write

${\displaystyle {\Bigl .}u(x)v(x){\Bigr |}_{a}^{b}=\int _{a}^{b}v\;du+\int _{a}^{b}u\;dv.}$

This expression may be rearranged to yield the familiar form of integration by parts

${\displaystyle \int _{a}^{b}u\;dv={\Bigl .}u(x)v(x){\Bigr |}_{a}^{b}-\int _{a}^{b}v\;du.}$

A generalization of integration by parts to higher dimensions is the Divergence Theorem also known as Gauss's Law, particularly when applied in electromagnetism.

## References

1. Greenspan, Harvey Philip, and David J. Benney. Calculus: an introduction to applied mathematics. H, P. GREENSPAN, 1997.