What is the method for solving the Gaussian integral?

[Noorac]

Homework Statement

Find the Gaussian integral:

[itex]I = \int_{-\infty}^{\infty} e^{-x^2-4x-1}dx [/itex]

(That's all the information the task gives me, minus the [itex]I=[/itex], I just put it there to more easily show what I have tried to do)

2. The attempt at a solution
I tried to square [itex]I[/itex] and get a double integral:


[itex]I^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{(-x^2-4x-1)+(-y^2-4y-1)}dxdy [/itex]

and then my plan was to convert to polar-coordinates, however, this is my first time ever with double-integrals and/or switching to polarcoordinates, and I am kind of lost because every single example on the internet use the standard [itex]e^{-x^2}[/itex] gaussian function(and it is easy to see [itex]r^2=x^2+y^2[/itex]). Anyone who can push me in the right direction(I'm not even sure what finding the Gassuian integral means(?))?

 

[Dick]

Complete the square. -x^2-4*x-1=(-(x+2)^2+3). Try using that.

 

[Noorac]

Tried that earlier on, but didn't get anywhere with it. Been trying it some more now, but I still don't see it though. I don't see the next step, I'm going to try some more though=) Thanks

 

[Dick]

Tried that earlier on, but didn't get anywhere with it. Been trying it some more now, but I still don't see it though. I don't see the next step, I'm going to try some more though=) Thanks

Next step would be a change of variables, u=x+2. Keep thinking about it.

 

[Noorac]

There! At least I got to the same answer as Wolfram Alpha;

[itex] I = \sqrt{\pi}e^{3}[/itex]

I hope it is correct. The steps I did after changeing variables [itex]r^2=(x+2)^2 + (y+2)^2[/itex] was substituting

[itex]u=r^2[/itex]

[itex]\frac{du}{dr}= 2r[/itex]

[itex]du = 2rdr[/itex]

[itex]dr = \frac{du}{2r}[/itex]

And just left the 6-constant alone all until I did the actual integral:

[itex]2\pi \int_{0}^{\infty} re^{-u+6} \frac{du}{2r}[/itex]
→
[itex]\pi \int_{0}^{\infty} e^{-u+6}du = \pi e^6[/itex]

(I think it's correct)
Thanks for the help=)

 

[HACR]

wow that's great.

 

[Dick]

(I think it's correct)
Thanks for the help=)

That is great. But you don't have the repeat the polar coordinate trick every time you see a Gaussian integral. After you've done it once, you should just remember (or look up) [itex]\int_{-\infty}^{\infty} e^{-u^2} du=\sqrt{\pi}[/itex]

 

[Noorac]

Yeah, the next task was somewhat similar, same objective, and it took only 3 minutes compared to the 3-4 hours of the last one =)

Now it's on to triple-integrals and what will probably be the most fun weekend since school ended before christmas!

Again, thanks=)