Integration by parts

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 $F(x)={\frac {df}{dx}}$ on the closed interval $[a,b]$ as

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

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 $u(x),v(x)$, then we may write by the product rule

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

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

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

which may be written more simply as

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

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

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

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

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

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