Hello everyone, This is my first post. I would like to understand better the idea of differentiation under the integral sign. I read about it in http://mathworld.wolfram.com/LeibnizIntegralRule.html and Feynman's autobiography, about evaluating an integral by differentiation under the integral sign, but how exactly it is done. Thank to everyone.
How it is done Consider [tex]I(b)=\int_0^1 \frac{x^b-1}{lnx} dx[/tex] now u can see clearly that after plugging the limits the variable x will vanish the only variable remains is b so the integration will be a function with b While integrating w.r.t x u consider b as a constant similarly when differentiating w.r.t b u consider x as a constant So , u have [tex]I'(b)=\int_0^1 \frac{x^b lnx}{lnx} dx[/tex] [tex]I'(b)=\int_0^1 x^b dx=\frac{1}{b+1}[/tex] [tex]=> I(b)= \int \frac{1}{b+1} db +c[/tex] If b=0 I(b)=0 => c=0 Therefore I(b)=ln(b+1) So clearly it is afunction of b now with no x
No, Feynman basically says that his "mathematical toolbox" (which included differentiation under the integral sign) was different from others', so he could solve problems others couldn't...