Efficiently Solve Gaussian Integration with Expert Homework Help

[TheMightyJ]

Homework Statement

Basically, i have to find the solution to:

Int( x² * exp (-(x-w)^2) , x= -infinity.. infinity)

Homework Equations

I realize this is connected to Gaussian Integration. So that if i have Int (exp(-x²), x=-infinity ... infinity) the answer is sqrt(Pi)

Also, i have read that there is a trick to solving such an integral.

you would consider F(a) = Int exp( - a * x^2) ( = sqrt(pi/a) ).

and then int dx x^2 exp(-x^2) = - F'(a) for a=1.

i understand this.

The Attempt at a Solution

Basically, the above is how i have attempted to work a way towards the solution, the trouble i am having is with a substitution i realize i must make at some point. at some point i must have x = x - w

But how do i implement this substitution??


This is my first post here, so hopefully that was somewhat clear and the relevant info is there. Thanks for any help.

 

[Nick89]

I don't know if this is the easiest solution, but you can try u = x - w. The integral splits into three integrals, which can be solved easily.

u = x-w
du = dx

Then, x = u+w, so x^2 = (u+w)^2, and the integral becomes:
$$\int (u+w)^2 e^{-u^2}\,du$$

 

[TheMightyJ]

Will give that a try! thank you very much!

 

[TheMightyJ]

Yep, tried it and it worked, brilliant, thank you!

 

[TheMightyJ]

Okay, so there is another question, similar in some ways to the previous one, but i am also having trouble. Mainly I am stuck on where to start!

The integral this time is

Int ( e^{-(x-w)2 - j*x} dx

the limits of integration are again, -infinity to infinity.

obviously there is some sort of gaussian integral stuff going on, but that "-j*x" has thrown me off, how should i deal with this?

just an idea to help me get started would be muchly appreciated, thanks!

 

[Cyosis]

You need to complete the square in this case. That is write (x-w)^2+jx as (...)^2+constant.

 

[TheMightyJ]

Obviously. Thank you! really not spotting obvious techniques today. Thanks for the help!

 

[Nick89]

j is not the imaginary number j (or i) right? If it is, that makes matters a little different.

 

[TheMightyJ]

no no, j is just the coefficient of x. Thanks.