Question: Compute the Fourier transform of the signal
$ x(t) = e^{-5t}u(t) + e^{-4(t-1)}u(t-3)\! $
Answer:
$ X(w) = \int_{-\infty}^{\infty}x(t)e^{-jwt}dt $
$ =\int_{-\infty}^{\infty}(e^{-5t}u(t) + e^{-4(t-1)}u(t-3))e^{-jwt}dt $
$ =\int_{0}^{\infty}e^{-5t} e^{-jwt}dt + \int_{3}^{\infty}e^{-4(t-1)}e^{-jwt}dt $
$ =\frac{e^{-(5+jw)t}}{-(5+jw)}\bigg]^{\infty}_{0}+ \int_{3}^{\infty}e^{-4t}e^{4}e^{-jwt}dt $
$ =\frac{e^{-5t}e^{-jwt}}{-(5+jw)}\bigg]^{infty}_{0} + e^4*\int_{3}^{\infty}e^{-(4+jw)t}dt $
$ =\frac{e^{-5t}e^{-jwt}}{-(5+jw)}\bigg]^{infty}_{0} + e^4 * \frac{e^{-(4+jw)t}}{-(4+jw)}\bigg]_3^\infty $
$ =\frac{1}{(5+jw)}+ e^4 * \frac{e^{-(12+3jw)}}{(4+jw)} $
$ =\frac{1}{(5+jw)} + \frac{e^{-(8+3jw)}}{(4+jw)} $
