# 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]

$\displaystyle \int_a^b F(x) \; dx = \int_a^b \frac{d f(x) }{dx} \; dx = f(b) - f(a)$

## 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

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

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

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

which may be written more simply as

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

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

$\displaystyle \Bigl. u(x)v(x) \Bigr|_a^b = \int_a^b v \; d u + \int_a^b u \; dv .$

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

$\displaystyle \int_a^b u \; dv = \Bigl. u(x)v(x) \Bigr|_a^b - \int_a^b v \; d u .$

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.