How to calculate Fourier Transform of e^-a*|t|?

1. The problem statement, all variables and given/known data

Calculate (from the definition, no tables allowed) the Fourier Transform of [itex]e^{-a*|t|}[/itex], where a > 0.



2. Relevant equations



Fourier Transform:



[itex]G(f) = \int_{-\infty}^{\infty} g(t)e^{-j\omega t} dt[/itex]



3. The attempt at a solution



I thought I'd break up the problem into the two cases of t (where it's negative and positive). However, when I calculated the portion where t > 0, I got:



[itex]

G(f) = \int_{0}^{\infty} e^{-at} e^{-j\omega t}dt = G(f) = \int_{0}^{\infty} e^{-(j\omega + a)t}dt = \frac{e^{-(j\omega + a)t}}{-(j\omega + a)}\bigg|_0^\infty = 0 - \frac{1}{-(j\omega + a)} = \frac{1}{j\omega + a}

[/itex]



Which is nowhere close to what WolframAlpha says the answer should be:



http://ift.tt/1mrJrHi



So I guess I'm confused on how I should even approach the problem. Any suggestions?