An integration method

[Saint Medici]

I was flipping through "Surely you're joking, Mr. Feynman" and I came across something he said that I'm curious about. I'll go ahead and quote it:

"The book showed how to differentiate parameters under the integral sign - it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn too again and again."

My question is, what method is he referring to? I'm only in vector cal, so I don't know if it's a method that is associated with higher-level mathematics, or if it's just something that I've "learned" and forgotten or what. So if anyone could enlighten me as to this method, how it's done, when it's used, etc., I'd be much appreciative. Thanks.

 

[mathman]

Your quote refers to differentiating parameters under the integral sign. I am not sure what else could be meant.

 

[HallsofIvy]

The only thing I could think of was "Leibniz's rule" which certainly is taught, sometimes in both advanced Calculus and Differential Equations courses (where it is used extensively):
$$\frac{\partial}{\partial x}\int_{\alpha(x)}^{\beta(x)}f(x,t)dt= \int_{\alpha(x)}^{\beta(x)}\frac{\partial f(x,t)}{\partial x}dt + \frac{d\alpha(x)}{dx}f(x,\alpha(x))- \frac{d\beta(x)}{dx}f(x,\beta(x))$$

(thanks, arildo!)

Alright, already! Is it good now? You know I can't be worried about little thing like one more or less "dt". (And "Liebniz" and "Lagrange" were really the same guy weren't they!)

 

[arildno]

"Lagrange's rule"?
That's odd; I know it as "Leibniz' rule"..

 

[HallsofIvy]

Wow, you're fast! I hadn't finished editing!

 

[arildno]

Well, it could just be one rule you were referring to, whatever shape you first presented it in..
Note:
You have a sign flaw in the upper&lower limit differentiations.

 

[Hurkyl]

And there should be a dt in there somewhere...

And to be fully general there needs to be limits sprinkled into there somehow...

 

[arildno]

The only thing I could think of was "Leibniz's rule" which certainly is taught, sometimes in both advanced Calculus and Differential Equations courses (where it is used extensively):
$$\frac{\partial}{\partial x}\int_{\alpha(x)}^{\beta(x)}f(x,t)dt= \int_{\alpha(x)}^{\beta(x)}\frac{\partial f(x,t)}{\partial x}dt + \frac{d\alpha(x)}{dx}f(x,\alpha(x))- \frac{d\beta(x)}{dx}f(x,\beta(x))$$

(thanks, arildo!)

Alright, already! Is it good now? You know I can't be worried about little thing like one more or less "dt". (And "Liebniz" and "Lagrange" were really the same guy weren't they!)

I (almost..) hate to be picky, but I prefer it this way:
$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)}f(x,t)dt= \int_{\alpha(x)}^{\beta(x)}\frac{\partial f(x,t)}{\partial x}dt + \frac{d\beta(x)}{dx}f(x,\beta(x))- \frac{d\alpha(x)}{dx}f(x,\alpha(x))$$