# Integration by parts

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\\[6pt]&=u(b)v(b)-u(a)v(a)-\int _{a}^{b}u'(x)v(x)\,dx.\end{aligned}}}

Or, letting ${\displaystyle u=u(x)}$ and ${\displaystyle du=u'(x)\,dx}$ while ${\displaystyle v=v(x)}$ and ${\displaystyle dv=v'(x)\,dx}$, the formula can be written more compactly:

${\displaystyle \int u\,dv\ =\ uv-\int v\,du.}$

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.