Integration by parts is a technique for performing indefinite integration or definite
integration by expanding the differential
of a product of functions and expressing
the original integral in terms of a known integral . A single
integration by parts starts with
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(1)
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and integrates both sides,
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(2)
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Rearranging gives
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(3)
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For example, consider the integral and let
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(4)
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(5)
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so integration by parts gives
where is a constant of integration.
The procedure does not always succeed, since some choices of may lead to more
complicated integrals than the original. For example, consider again the integral
and let
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giving
which is more difficult than the original (Apostol 1967, pp. 218-219).
Integration by parts may also fail because it leads back to the original integral. For example, consider and
let
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(11)
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then
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(12)
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which is same integral as the original (Apostol 1967, p. 219).
The analogous procedure works for definite integration by parts, so
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(13)
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where .
Integration by parts can also be applied times to :
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(14)
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Therefore,
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(15)
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But
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(16)
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(17)
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so
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(18)
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Now consider this in the slightly different form . Integrate
by parts a first time
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(19)
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so
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(20)
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Now integrate by parts a second time,
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(21)
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so
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(22)
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Repeating a third time,
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(23)
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Therefore, after applications,
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(24)
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If (e.g., for an th degree polynomial), the last term is 0, so the sum terminates after
terms and
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(25)
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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 12, 1972.
Apostol, T. M. "Integration by Parts." §5.9 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an
Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 217-220,
1967.
Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag,
p. 269, 1997.
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