Calculating the Fourier transform of shifted scaled unit step function
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.