Solution
= Solution
Split the integral at the origin and reflect the negative half:
$$
\begin{aligned}
\widehat{e^{-a|x|}}(s)
&=\int_0^\infty e^{-(a+is)x}\,dx+\int_0^\infty e^{-(a-is)x}\,dx\\
&=\frac1{a+is}+\frac1{a-is}=\frac{2a}{a^2+s^2}.
\end{aligned}
$$
Thus the <Fourier transform of a two-sided exponential> is
$$
\boxed{\widehat{e^{-a|x|}}(s)=\frac{2a}{a^2+s^2}\quad(a>0).}
$$