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单词 ExampleOfDifferentiationUnderIntegralSign
释义

example of differentiation under integral sign


DifferentiationMathworldPlanetmath with respect to a parameter under the integral sign may sometimes yield useful formulae.  One example is given here.

We know that the equation

01xm𝑑x=1m+1

is valid for all  m>-1.  If one differentiates with respect to m under the integral sign (http://planetmath.org/DifferentiationUnderTheIntegralSign) in succession, one gets

01memlnx𝑑x=01emlnxlnxdx=01xmlnxdx=-1(m+1)2
01mxmlnxdx=01xm(lnx)2𝑑x=+12(m+1)3
01mxm(lnx)2𝑑x=01xm(lnx)3𝑑x=-123(m+1)4

It’s evident that repeating the differentiation n times the final result is the

01xm(lnx)ndx=(-1)nn!(m+1)n+1  (m>-1).
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更新时间:2025/5/3 18:58:46