Example 1
Compute the Fourier Transform of $ x(t)=e^{-t}u(t) $.
$ X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $
$ =\int_{-\infty}^{\infty}e^{-t}u(t)e^{-j\omega t}dt $
$ =\int_{0}^{\infty}e^{-t}e^{-j\omega t}dt $
$ =\int_{0}^{\infty}e^{-(1+j\omega )t}dt $
$ =[\frac {e^{-(1+j\omega )t}}{-(1+j\omega)}]|_0^\infty $
$ =\frac {e^{-(1+j\omega )\infty}}{-(1+j\omega)}-\frac {e^{-(1+j\omega )0}}{-(1+j\omega)} $
$ =0+\frac {1}{(1+j\omega)} $
$ =\frac {1}{(1+j\omega)} $
