$ x(t) = e^{-|t-1|} \, $
$ X(w) = \int_{-\infty}^{\infty}e^{-|t-1|}e^{-jwt}dt $
$ X(w) = \int_{-\infty}^{1}e^{(t-1)}e^{-jwt}dt+\int_{1}^{\infty}e^{-(t-1)}e^{-jwt}dt $
$ X(w) = \int_{-\infty}^{1}e^{-1}e^{(1-jw)t}dt+\int_{1}^{\infty}e^{1}e^{-(1+jw)t}dt $
$ X(w) = {\left.\frac{e^{-1}e^{(1-jw)t}}{1-jw}\right]^1_{-\infty} }+{\left.\frac{-e^{1}e^{-(1+jw)t}}{1+jw}\right]^{\infty}_1 } $$ = e^{-1}\frac{e^{(1-jw)}}{1-jw}+e^{1}\frac{e^{(-(1+jw)}}{1+jw} $
$ X(w) = \frac{e^{-jw}}{1-jw}+\frac{e^{-jw}}{1+jw} $
$ X(w) = \frac{2e^{-jw}}{1+w^2} $
