[tex]\int_{-3}^3 |t| e^{-j \omega t} \, dt = \int_{0}^3 t e^{-j \omega t} \, dt - \int_{-3}^0 t e^{-j \omega t} \, dt[/tex]
and then you can solve both integrals with a trick (write the integrand as a derivative w.r.t. omega, for example).
Alternatively, you can use Euler's identity to write the integral as
[tex]\int_{-3}^3 |t| \cos(\omega t) \, dt + j \int_{-3}^3 |t| \sin(\omega t) \, dt[/tex]
and use (anti)-symmetries to reduce the problem before taking care of the absolute value.