Differentiating Under the Integral Sign

Date: 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/