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Concept
Integration by parts
Summary
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:
Or, letting and while and the formula can be written more compactly:
Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. 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.
The theorem can be derived as follows. For two continuously differentiable functions u(x) and v(x), the product rule states:
Integrating both sides with respect to x,
and noting that an indefinite integral is an antiderivative gives
where we neglect writing the constant of integration. This yields the formula for integration by parts:
or in terms of the differentials ,
This is to be understood as an equality of functions with an unspecified constant added to each side. Taking the difference of each side between two values x = a and x = b and applying the fundamental theorem of calculus gives the definite integral version:
The original integral ∫ uv′ dx contains the derivative v′; to apply the theorem, one must find v, the antiderivative of v', then evaluate the resulting integral ∫ vu′ dx.
It is not necessary for u and v to be continuously differentiable. Integration by parts works if u is absolutely continuous and the function designated v′ is Lebesgue integrable (but not necessarily continuous). (If v′ has a point of discontinuity then its antiderivative v may not have a derivative at that point.
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