Transcription of DIFFERENTIATING UNDER THE INTEGRAL SIGN
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DIFFERENTIATING UNDER THE INTEGRAL SIGN
DIFFERENTIATING UNDER THE INTEGRAL SIGNKEITH CONRADI had learned to do integrals by various methods shown in a book that my highschool physics teacher Mr. Bader had given me. [It] showed how to differentiateparameters UNDER the INTEGRAL sign it s a certain operation. It turns out that snot taught very much in the universities; they don t emphasize it. But I caught onhow to use that method, and I used that one damn tool again and again. [If] guysat MIT or Princeton had trouble doing a certain INTEGRAL , [then] I come along andtry DIFFERENTIATING UNDER the INTEGRAL sign , and often it worked. So I got a greatreputation for doing integrals, only because my box of tools was different fromeverybody else s, and they had tried all their tools on it before giving the problemto Feynman [5, pp. 71 72] method of differentiation UNDER the INTEGRAL sign , due to Leibniz in 1697 [4], concerns integralsdepending on a parameter, such as 10x2e txdx.
DIFFERENTIATING UNDER THE INTEGRAL SIGN 5 Call the integral on the left I. For t2R, set F(t) = Z 1 0 e t 2(1+x )=2 1 + x2 dx: Then F(0) = R 1 0 dx=(1 + x2) = ˇ=2 and F(1) = 0.Di erentiating under the integral sign,
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