| Thread Closed |
Differentiation under the integral sign |
Share Thread | Thread Tools |
| Jan21-04, 05:20 PM | #1 |
|
|
Differentiation under the integral sign
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. |
| Jan21-04, 11:34 PM | #2 |
|
|
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 |
| Jan22-04, 09:00 PM | #3 |
|
|
Funny, I was just reading Surely You're Joking, Mr. Feynman and I was wondering about that also.
|
| Jan23-04, 09:01 AM | #4 |
|
|
Differentiation under the integral sign |
| Jan23-04, 09:09 AM | #5 |
|
|
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...
|
| Thread Closed |
| Thread Tools | |
Similar Threads for: Differentiation under the integral sign
|
||||
| Thread | Forum | Replies | ||
| 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 | ||