Register to reply

Differentiation under the integral sign

by hliu8
Tags: differentiation, integral, sign
Share this thread:
Jan21-04, 05:20 PM
P: 3
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 and Feynman's autobiography, about evaluating an integral by differentiation under the integral sign, but how exactly it is done.

Thank to everyone.
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Jan21-04, 11:34 PM
himanshu121's Avatar
P: 658
How it is done

[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
Jan22-04, 09:00 PM
P: 45
Funny, I was just reading Surely You're Joking, Mr. Feynman and I was wondering about that also.

Jan23-04, 09:01 AM
P: 3,243
Differentiation under the integral sign

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?
Jan23-04, 09:09 AM
P: 695
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...

Register to reply

Related Discussions
Differentiating under the integral sign Calculus 13
Differentiation under the integral Calculus & Beyond Homework 5
Differentiation of Integral Calculus & Beyond Homework 18
Integral sign error Calculus & Beyond Homework 3
Using differentiation under integral sign to proof equality of mixed partials Calculus 2