Motivation of the Gaussian Integral I read on Wikipedia that Laplace was the first to evaluate
$$\int\nolimits_{-\infty}^\infty e^{-x^2} \, \mathrm dx$$
Does anybody know what he was doing that lead him to that integral? Even better, can someone pose a natural problem that would lead to this integral?
Edit: Many of the answers make a connection to the normal distribution, but then the question now becomes: Where does the density function of the normal distribution come from? Mike Spivey's answer is in the spirit of what I am looking for: an explanation that a calculus student might understand.

The integral you gave, when taken as a definite integral :
$\int^{x_2}_{x_1} e^{-x^2} dx$
When scaled by $\frac {1}{\pi^{0.5}}$
is/describes the univariate probability density of a normally-distributed trandom variable $X$ with mean=0 and standard deviation ${\frac {1}{2^{0.5}}}$, i.e.  This means that the numerical value of this integral gives you the probability of the event: $x_1 \leq X\leq x_2$
When this integral is scaled by the right factor $K$ 
 it describes a family of normal distributions with mean $\mu$ and standard deviation $\sigma$
You can show it integrates to that constant K (so that when you divide by $K$ , the value of the integral is $1$, which is what makes it into a density function) by using this trick (used for the case mean=0)
Set I=$C\int e^{-x^2}$  , then consider $\int e^{-y^2}$ , and then compute their product as (using the fact that $x^2$ is a constant when considered as a function of y, and viceversa for x ):
$I^2$=$\int e^{-x^2+y^2}dxdy$ , using a polar change of variable: $x^2+y^2=r^2$ (and, of course, a change of the regions of integration.)
The integral is based on non-mathematical assumptions too:
http://www.stat.tamu.edu/~genton/2007.AG.Bernoulli.pdf
