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The integral

This is one of the classical difficult integrals. This method is, I think, attributed to Gauss. The problem is that you cannot find a primitive function to the integrand, no matter what you do. So what to do? The trick is to realize that an infinite rectangle is the same as an infinite square. So we start by making the integral into two integrals, one in the *x*– and the *y*-direction.

Since the integral with respect to *y* can be seen as a constant as seen from the variable *x*, then we can move the whole integral inside the other. I.e.

This can be rewritten as

This is now an integral over a function

as seen in the figure.

As seen from above we can now integrate the whole area by working with small areas *ΔA=ΔxΔy*. We will now change to polar coordinates. We have that

For our *ΔA* we will use *ΔA=rΔθΔr* as seen in the figure.

So to integrate over the whole surface instead of integration from minus to plus infinity over both *x* and *y* we will integrate for a radius from 0 to infinity and an angle from 0 to 2*π*.

Putting it all together we get

Now, since the variables are independent we can do the opposite we did above when we put the integrals inside each other. We get

For the first part, we get

For the second part, we have

To find this integral we can do an ordinal variable substation:

This gives us

Combining these we get

and thus finally

We yet get another case where *π* and e are directly connected.

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