Let $ x(t) = e^{-at} u(t) $
$ \chi(w) = \mathcal{F} (x(t)) = \int^{\infty}_{-\infty} e^{-at} u(t) e^{-jwt} dt $
$ = \int^{\infty}_{0} e^{-at}.e^{-jwt} dt $
$ = \int^{\infty}_{0}e^{-(a+jw)t} dt $
$ = -\frac{1}{a+jw} [e^{-(a+jw)t}]^{\infty}_{0} $
$ = -\frac{1}{a+jw} [-1] $
$ =\frac{1}{a+jw} $
