In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.
The integration by parts formula states:
{\displaystyle {\begin{aligned}\int {a}^{b}u(x)v’(x),dx&={\Big [}u(x)v(x){\Big ]}{a}^{b}-\int _{a}^{b}u’(x)v(x),dx\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u’(x)v(x),dx.\end{aligned}}}{\displaystyle {\begin{aligned}\int {a}^{b}u(x)v’(x),dx&={\Big [}u(x)v(x){\Big ]}{a}^{b}-\int _{a}^{b}u’(x)v(x),dx\&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u’(x)v(x),dx.\end{aligned}}}
Or, letting {\displaystyle u=u(x)}{\displaystyle u=u(x)} and {\displaystyle du=u’(x),dx}{\displaystyle du=u’(x),dx} while {\displaystyle v=v(x)}{\displaystyle v=v(x)} and {\displaystyle dv=v’(x),dx}{\displaystyle dv=v’(x),dx}, the formula can be written more compactly:
{\displaystyle \int u,dv\ =\ uv-\int v,du.}{\displaystyle \int u,dv\ =\ uv-\int v,du.}
Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.[1][2] More general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts.
Contents
1 Theorem
1.1 Product of two functions
1.2 Validity for less smooth functions
1.3 Product of many functions
2 Visualization
3 Applications
3.1 Finding antiderivatives
3.1.1 Polynomials and trigonometric functions
3.1.2 Exponentials and trigonometric functions
3.1.3 Functions multiplied by unity
3.1.4 LIATE rule
3.2 Wallis product
3.3 Gamma function identity
3.4 Use in harmonic analysis
3.4.1 Fourier transform of derivative
3.4.2 Decay of Fourier transform
3.5 Use in operator theory
3.6 Other applications
4 Repeated integration by parts
4.1 Tabular integration by parts
5 Higher dimensions
5.1 Green’s first identity
6 See also