The other day i was wondering in my computer about functions who would look like a gaussian distribution. I first tought of the function: $$ f(x) = \frac{1}{x^2 + 1} $$ Whose max value is $f(0) = 1$. However, we have that $$ \int_{-\infty}^{\infty} \frac{1}{x^2 + 1} \mathbb{d}x = \pi $$ And hence this cannot be a probability distribution. So basically i kept trying to find a constant $c^2$ such that the integral would evaluate $1$. And, for my (at first glance) surprise: $$ \int_{-\infty}^{\infty} \frac{1}{x^2 + \pi^2} \mathbb{d}x = 1 $$ (Since the primitive of $f(x)$ is $\operatorname{arctan}x$, this is not very surprising). I found wonderful that $\pi$ appeared in such an unexpected way.
My question is: Does this probability distribution has any relevance like the other we normally use in the study of probability? (Gaussian, Exponential, Poisson, etc) For me it seems like such a fine, simple probability distribution. Why haven't i ever seen it anywhere?