Let $X_1, \dots, X_n$ be independent random variables with common density function of the form $$f(x) = \frac{c}{1+x^2}$$
Show that $\frac 1n (X_1+\dots +X_n)$ has the same law as $X_1$. The hint is to use Levy's theorem for characteristic functions. However, I can't find the characteristic function of $X_1$. I set up the integral $$\varphi_X(t) = \int_{\mathbb R}e^{itx}\frac{c}{1+x^2} dx$$ but I can't seem to figure out how to solve it.
HINT
This is a Cauchy distribution with parameters $x_0=0$ and $\gamma=1$, also $c=\frac 1{\pi}$. The characteristic function is known to be
$$e^{-\mid t\mid}.$$