Solution to the Gaussian integral Exp(-x^2) with explanation

One of the most frequently found integrals in quantum mechanics and many other branches of physics is a variation of the Gaussian integral

where the answer is an unknown result we will call I. So the integral will look like this.

Now we must remember that we can also write I like this replacing x with y since x and y have the same value here.

If we multiply I by itself the result we get is this

From here we can combine these two integrals into one integral dependent on both x and y like I do here.

Now is where we have to get a little creative. One thing we can do is switch coordinates from Cartesian to polar coordinates since we know that

and r can be inserted into the integral for x and y. The limits will also be changed since we are now in polar coordinates, we will be integrating from zero to infinity and from zero to two pi since we are integrating over a circle. The integral now looks like this

integrating over theta is simple and is simply two pi, leaving the r integral to be evaluated.

Now it is time to use u substitution. We will use u is equal to r squared and insert into the integral.

Finding du and using a bit of algebra we quickly get

and insert into the integral

As you can see, the two and the r quickly cancel, leaving

Using the properties of integrating exponential functions we can quickly evaluate this integral and take the limits

Inserting o and infinity into the exponential function and taking the differences gives us

The exponential of infinity is zero, and the exponential of zero is one. The negative going into a negative gives us a positive one

The answer is now obvious, taking the square root of each side, we get

so to wrap things up, the result of the Gaussian integral is

This integral and variations are used so frequently in quantum mechanics and statistical mechanics that I felt it might be useful to go over the solution here. Hopefully this helps someone!