Leibniz Integral Rule

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

(1)

It is sometimes known as differentiation under the integral sign.

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

			(2)
			(3)

for  (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."

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.