Integration by parts

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Integration by Parts (Partial integration)

Integration by parts (or partial integration) is one of the most useful basic calculus operations. It is a generalization of the fundamental theorem of calculus for the case of the product of two functions.

Fundamental theorem of Calculus

The classical definition of the definite integral of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x) = \frac{df}{dx} } on the closed interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [a,b] } as [1]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_a^b F(x) \; dx = \int_a^b \frac{d f(x) }{dx} \; dx = f(b) - f(a) }

Integration by parts

The second important result from calculus is the form of the Fundamental Theorem known as integration by parts or partial integration. If we consider two differentiable functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x), v(x) } , then we may write by the product rule

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} \left( u(x) v(x) \right) = \frac{d u}{dx} v(x) + u(x) \frac{dv}{dx} }

Integrating both sides from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b } yields


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_a^b \left[ \frac{d}{dx} \left( u(x) v(x) \right) \right] \; dx = \int_a^b \left[ \frac{d u}{dx} v(x) + u(x) \frac{dv}{dx} \right] \, dx }

which may be written more simply as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_a^b \left[ \frac{d}{dx} \left( u(x) v(x) \right) \right] \; dx = \int_a^b v \; d u + \int_a^b u \; dv . }

Recognizing that the term on the left is an exact differential, we may write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Bigl. u(x)v(x) \Bigr|_a^b = \int_a^b v \; d u + \int_a^b u \; dv . }

This expression may be rearranged to yield the familiar form of integration by parts

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_a^b u \; dv = \Bigl. u(x)v(x) \Bigr|_a^b - \int_a^b v \; d u . }

A generalization of integration by parts to higher dimensions is the Divergence Theorem also known as Gauss's Law, particularly when applied in electromagnetism.

References

  1. Greenspan, Harvey Philip, and David J. Benney. Calculus: an introduction to applied mathematics. H, P. GREENSPAN, 1997.