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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
(1)

and integrates both sides,
(2)

Rearranging gives
(3)

For example, consider the integral and let
(4)
 
(5)

so integration by parts gives
where is a constant of integration.
The procedure does not always succeed, since some choices of may lead lớn more complicated integrals phàn nàn the original. For example, consider again the integral and let
(8)

giving
which is more difficult phàn nàn the original (Apostol 1967, pp. 218219).
Integration by parts may also fail because it leads back lớn the original integral. For example, consider and let
(11)

then
(12)

which is same integral as the original (Apostol 1967, p. 219).
The analogous procedure works for definite integration by parts, so
(13)

where .
Integration by parts can also be applied times lớn :
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Therefore,
(15)

But
(16)

(17)

so
(18)

Now consider this in the slightly different khuông . Integrate by parts a first time
(19)

so
(20)

Now integrate by parts a second time,
(21)

so
(22)

Repeating a third time,
(23)

Therefore, after applications,
(24)

If (e.g., for an th degree polynomial), the last term is 0, ví the sum terminates after terms and
(25)

See also
Darboux's Formula, Integral, Integration, Inverse Function Integration, Summation by Parts
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References
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: OneVariable Calculus, with an Introduction lớn Linear Algebra. Waltham, MA: Blaisdell, pp. 217220, 1967.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: SpringerVerlag, p. 269, 1997.
Cite this as:
Weisstein, Eric W. "Integration by Parts." From MathWorldA Wolfram Web Resource. https://mamnonkidzone.edu.vn/IntegrationbyParts.html
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