To integrate the expression ∫_{-3}^{3}|t|e^{-jωt}dt, it is necessary to consider the absolute value, which can be addressed by splitting the integral into two parts: one from 0 to 3 and another from -3 to 0. This leads to the equation ∫_{-3}^{3}|t|e^{-jωt}dt = ∫_{0}^{3}te^{-jωt}dt - ∫_{-3}^{0}te^{-jωt}dt. Another approach involves using Euler's identity to separate the integral into real and imaginary components, resulting in ∫_{-3}^{3}|t|cos(ωt)dt + j∫_{-3}^{3}|t|sin(ωt)dt. Utilizing symmetries can simplify the calculations, particularly when addressing the absolute value. Both methods effectively tackle the integration of absolute values with complex exponentials.