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Leibniz Integral Rule


The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential variable,

 partial/(partialz)int_(a(z))^(b(z))f(x,z)dx=int_(a(z))^(b(z))(partialf)/(partialz)dx+f(b(z),z)(partialb)/(partialz)-f(a(z),z)(partiala)/(partialz).
(1)

It is sometimes known as differentiation under the integral sign.

This rule can be used to evaluate certain unusual definite integrals such as

phi(alpha)=int_0^piln(1-2alphacosx+alpha^2)dx
(2)
=2piln|alpha|
(3)

for |alpha|>1 (Woods 1926).

Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."


See also

Derivative, Integral, Integration Under the Integral Sign

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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. 11, 1972.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 232, 1987.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, pp. 20-21, 2004.Feynman, R. P. "A Different Set of Tools." In 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character. New York: W. W. Norton, 1997.Hijab, O. Introduction to Calculus and Classical Analysis. New York: Springer-Verlag, p. 189, 1997.Kaplan, W. "Integrals Depending on a Parameter--Leibnitz's Rule." §4.9 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 256-258, 1992.Woods, F. S. "Differentiation of a Definite Integral." §60 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 141-144, 1926.

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Leibniz Integral Rule

Cite this as:

Weisstein, Eric W. "Leibniz Integral Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizIntegralRule.html

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