Differentiation under the integral sign

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Discussion Overview

The discussion centers on the concept of differentiation under the integral sign, exploring its application and understanding through examples and references to literature, particularly Feynman's autobiography.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant seeks a deeper understanding of differentiation under the integral sign, referencing resources like MathWorld and Feynman's autobiography.
  • Another participant provides an example involving the integral I(b) = ∫(0 to 1) (x^b - 1) / ln(x) dx, explaining the process of differentiation with respect to b while treating x as a constant.
  • This participant claims to derive I(b) = ln(b + 1) through differentiation, although the steps involve assumptions about constants and integration.
  • Several participants express interest in Feynman's approach, with one asking for a technical explanation of what Feynman wrote about this method in his book.
  • Another participant notes that Feynman described his mathematical toolbox as unique, suggesting it allowed him to tackle problems differently than others.

Areas of Agreement / Disagreement

Participants express curiosity and share insights about the topic, but there is no consensus on the technical details or the interpretation of Feynman's approach. The discussion remains exploratory with varying levels of understanding and interest.

Contextual Notes

Some mathematical steps and assumptions in the differentiation process are not fully resolved, and the discussion reflects differing levels of familiarity with the topic.

hliu8
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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.
 
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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
 
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Funny, I was just reading Surely You're Joking, Mr. Feynman and I was wondering about that also.
 
Originally posted by Tron3k
Funny, I was just reading Surely You're Joking, Mr. Feynman and I was wondering about that also.
what is written about this in the book, is there a technical explanation about it?
 
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...
 
Last edited:

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