Editing Integration by parts (section)

{{Short description|Mathematical method in calculus}}
{{Calculus |Integral}}

In [[calculus]], and more generally in [[mathematical analysis]], '''integration by parts''' or '''partial integration''' is a process that finds the [[integral (mathematics)|integral]] of a [[product (mathematics)|product]] of [[Function (mathematics)|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 [[derivative|differentiation]]; it is indeed derived using the product rule.

The integration by parts formula states:
<math display="block">\begin{align}
   \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{align}</math>

Or, letting <math>u = u(x)</math> and <math>du = u'(x) \,dx</math> while <math>v = v(x)</math> and <math>dv = v'(x) \, dx,</math> the formula can be written more compactly:
<math display="block">\int u \, dv \ =\ uv - \int v \, du.</math>

The former expression is written as a definite integral and the latter is written as an indefinite integral. Applying the appropriate limits to the latter expression should yield the former, but the latter is not necessarily equivalent to the former.

Mathematician [[Brook Taylor]] discovered integration by parts, first publishing the idea in 1715.<ref name="Brook Taylor biography, St. Andrews">{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.html |title=Brook Taylor |work=History.MCS.St-Andrews.ac.uk |access-date= May 25, 2018}}</ref><ref name="Brook Taylor biography, Stetson">{{cite web |url=https://www2.stetson.edu/~efriedma/periodictable/html/Tl.html |title=Brook Taylor |work=Stetson.edu |access-date=May 25, 2018 |archive-date=January 3, 2018 |archive-url=https://web.archive.org/web/20180103003304/http://www2.stetson.edu/~efriedma/periodictable/html/Tl.html |url-status=dead }}</ref> More general formulations of integration by parts exist for the [[Riemann–Stieltjes integral#Properties and relation to the Riemann integral|Riemann–Stieltjes]] and [[Lebesgue–Stieltjes integral#Integration by parts|Lebesgue–Stieltjes integrals]]. The [[Discrete mathematics|discrete]] analogue for [[Sequence|sequences]] is called [[summation by parts]].