Integration by parts
Mathematical method in calculus / From Wikipedia, the free encyclopedia
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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.
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The integration by parts formula states:
Or, letting $u=u(x)$ and $du=u'(x)\,dx$ while $v=v(x)$ and $dv=v'(x)\,dx,$ the formula can be written more compactly:
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.