Characteristic function of $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$ Let X denote a real-valued random variable with an absolutely continuous distribution with density function  $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$.
Find the characteristic function.
Attempt:
I know that the setup should look something like this:
$$\frac{1}{2} \int_{-\infty}^\infty e^{itx} e^{-|x|} \;dx $$

The characteristic function is
$$\begin{align} \frac12 \int_{-\infty}^{\infty} e^{-|x|} e^{i t x} ~dx &= \frac12 \int_{-\infty}^0 e^{x+i t x} ~dx + \frac12 \int_0^{\infty} e^{-x + i t x} ~dx\\ &= \frac12 \int_0^{\infty} e^{-(1+i t) x}  ~dx + \frac12 \int_0^{\infty} e^{-(1-i t) x} ~dx\\ &= \frac12 \frac{1}{1+i t} + \frac12 \frac{1}{1-i t}\\ &= \frac{1}{1+t^2}\end{align}$$
