Prove that $ϕ_{X_1}(u) =e^{−|u|}$ is the characteristic function of $f_{X_{n}}(x) = \frac{1}{π(1+x^2)}$

Question

I have a problem with this exercise:

Let $(X_{n})_{n ∈ \mathbb{N}}$ be a sequence of independent, identically distributed, real-valued random variables whose distribution function is absolutely continuous with density: $f_{X_{n}}(x) = \frac{1}{π(1+x^2)}$.

I have to show that $ϕ_{X_1}(u) =e^{−|u|}, ∀u ∈ \mathbb{R}$ is the characteristic function of $f_{X_1}$

I know that a characteristic function is really the characteristic function of distribution fuction when it applies:

$f_X(x)=\frac{1}{2\pi}\int_{- \infty}^{\infty}e^{-itx}ϕ(t)dt$

So I have inserted what I have in the equation:

$f_X(x)=\frac{1}{2\pi}\int_{- \infty}^{\infty}e^{-itx}e^{-|t|}dt$

But now i don't know how to continue, can someone help me?