Example of Computation of Fourier transform of a CT SIGNAL

Specify a signal x(t) and compute its Fourier transform using the integral formula.( Make a hard one)

$e^{-2(t-1)}u(t-1)\,$

$\,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\,$

$\,\mathcal{X}(\omega)= \int_{1}^{ \infty} e^{2-t(2+jw)}dt\,$

integrating and putting in limits

$\,\mathcal{X}(\omega)= \frac{e^{2-(2+jw)}}{2+jw} \,$

$\,\mathcal{X}(\omega)= \frac{e^{2-2-jw}}{2+jw} \,$

$\,\mathcal{X}(\omega)= \frac{e^{-jw}}{2+jw} \,$