Calculating the Fourier transform of shifted scaled unit step function I have $x_1(t)$ here. To get $x_2(t)$, I need to differentiate $x_1(t)$. Express $x_2(t)$ as $2u(t+2)-4u(t)+2u(t-2)$.
From Fourier transform definition integral, I got $X_2(j\omega)=\frac{2e^{j\omega 2}}{j\omega}-\frac{4\pi \delta(\omega)}{j\omega}+\frac{2e^{-j\omega2}}{j\omega}$.
Is this correct? It seems weird and would be complicated to calculate the magnitude and phase spectrum. Thanks!
It looks like the idea of that exercise is to compute the Fourier transform of $x_3(t)$, and then from that derive the Fourier transform of the original function $x_1(t)$.

Note that $x_3(t)$ is just a sum of scaled and shifted Dirac impulses, so computing its Fourier transform is trivial. The Fourier transform of $x_2(t)$ is then easily derived by multiplying $X_3(j\omega)$ by $\pi\delta(\omega)+1/j\omega$, and noting that the Dirac delta is cancelled because $X_3(0)=0$. So you just end up dividing by $j\omega$. In the same way, $X_1(j\omega)$ is obtained from $X_2(j\omega)$. Also note that none of the three Fourier transforms $X_1(j\omega)$, $X_2(j\omega)$, and $X_3(j\omega)$ have any Dirac impulses.

It's a good idea to cross-check your result by realizing that $x_1(t)$ can also be represented as the convolution of two rectangular functions, i.e., its Fourier transform must be a squared sinc-function.