I got this function in time domain and I have to compute Fournier transform: $ x(t)=e^{-4|t|}$. My solution give me this help : $x(t)=e^{-a|t|} \Longleftrightarrow X(jw)=\frac{2a}{a^2+\omega^2}$. But from where this does come (except from my book)?
Thank you
$$X(jw) = \int_{-\infty}^{+\infty}e^{-a|t|}e^{-jwt}dt = \\ \int_{0}^{+\infty}e^{-at}e^{-jwt}dt + \int_{-\infty}^{0}e^{at}e^{-jwt}dt =\\ \left. \frac{e^{-(a+jw)t}}{-(a+jw)}\right|_0^{+\infty} + \left. \frac{e^{(a-jw)t}}{(a-jw)}\right|_{-\infty}^0 = \\ \frac{1}{a+jw} + \frac{1}{a-jw} = \frac{a+jw+a-jw}{(a+jw)(a-jw)} = \\\frac{2a}{a^2+w^2}. $$