Help With Integral Problem - Double Integration?

[Ami]

Homework Statement

http://www4.0zz0.com/2007/02/16/20/74624071.bmp

Homework Equations

Can anybody help me with this integral problem?Please.

The Attempt at a Solution

Is the only solution by the "douple integration" method ?

 

[NateTG]

What have you tried?

 

[drpizza]

Calc I or Calc II? Are you sure you wrote that problem down correctly?

(edit: have you heard of the imaginary error function? Either that, or I'm completely blanking out on this one.)

 

[Ami]

I'm really sorry. I've written the problem down uncorrectly

This is the right one:_
http://www4.0zz0.com/2007/02/16/21/95988123.bmp

sorry.And thanks for your replies.

 

[Hootenanny]

Try rewriting the problem;

$$\int e^{-x^2}\cdot e^{-\frac{1}{x^2}} dx$$

 

[Ami]

I tried this step,but I couldn't continue.

Thanks for your reply "Hootenanny"

 

[Ami]

Maybe we need "douple integration" here.

 

[lalbatros]

cheat with this site:


http://integrals.wolfram.com/index.jsp


enter your function as follows: Exp[-x^2-1/x^2]

You will see that it is not an easy one.

However, I guess that for a definite integral {-Inf,+Inf} it could be easier.
Are you sure you want the indefinite integral?

If yes, you can still try to figure out out to get by hand the result mathematica will show you.

The result is:

Integrate[Exp[-x^2 - 1/x^2], x] ==
(Sqrt[Pi]*((1 + Erf[-x^(-1) + x])/E^2 + E^2*(-1 + Erf[x^(-1) + x])))/4

 

[Ami]

Thank you very much for the wonderful site,and for replying.

I've tried it,and I saw the result. But as you said,I want to figure out how to get by hand the result.

I still need help.

 

[lalbatros]

I want to figure out how to get by hand the result

That's where cheating becomes really an art.
Try calculating the derivate of the result and write the calculation in reverse order.

In this process you may have some good idea.

Another method is to look in the Mathematica package how it does that.
It is very difficult to read, but it is easy to guess the ingredients.
Probably a standard form we be used and some integral table results (not many since erf is an easy function).
But you need Mathematica ... and that's very expensive

 

[Ami]

Great ideas,especially that one about cheating

I will try this one,because I can't use Mathematica package right now.
And we will see

Thanks for your great help.