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


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


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

∫01xm(lnx)ndx=(-1)n⁒n!(m+1)n+1  (m>-1).
Title example of differentiation under integral signMathworldPlanetmath
Canonical name ExampleOfDifferentiationUnderIntegralSign
Date of creation 2013-03-22 17:01:56
Last modified on 2013-03-22 17:01:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Example
Classification msc 26A24
Classification msc 26B15