Differentiating Under the Integral SignDate: 01/11/2001 at 15:26:38 From: Nicholas Mecholsky Subject: Differentiating under the integral Dr. Math, I have recently been looking for a resource that will explain the integration method of differentiating parameters under the integral sign. I have not heard of this method before, but I read a book Feynmann wrote and he couldn't stop talking about it. I thought it would be a useful method to learn. If you have any information as to where I could read about it, I would greatly appreciate it. Thank you for your time. Nick Date: 01/11/2001 at 16:15:14 From: Doctor Schwa Subject: Re: Differentiating under the integral Hi Nick, I tried finding a good Web site to explain it to you, but I couldn't. Here's an example of how to do it: Suppose you know the integral from -infinity to infinity of e^(-ax^2)dx is sqrt(pi/a): oo INT [e^(-ax^2) dx] = sqrt(pi/a) -oo Now, if you want to integrate things like xe^(-ax^2) it's not too hard; let u = ax^2 and so on. In fact, any odd power of x is okay. But what about those pesky even powers? The trick is to differentiate with respect to a, then do the integral with respect to x, then integrate with respect to a. That's called differentiating under the integral sign. In this example, the derivative of e^(-ax^2) is just: -x^2 e^(-ax^2) (Remember, we're now pretending that a is a variable and x is a constant.) So, from the equation above, taking the derivative of both sides with respect to a gives: oo -INT [x^2 e^(-ax^2) dx] = -1/2 sqrt(pi/a^3) -oo which is a lot easier than any other method of solving this infinite integral. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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