How do you find the Characteristic Function of a Standard Laplace Distribution:

Let X denote a real-valued random variable with an absolutely continuous distribution with density function p(x) = $\displaystyle \frac{1}{2}$$\displaystyle e^{-|x|}$, $\displaystyle \hspace{5mm}$-$\displaystyle \infty $< x < $\displaystyle \infty$.

Find the characteristic function.

Attempt:

I know that the setup should look something like this:

$\displaystyle \frac{1}{2}$$\displaystyle \int_{-\infty}^\infty$$\displaystyle e^{itx}$$\displaystyle e^{-|x|}$dx