How Do You Integrate Absolute Values with Complex Exponentials?

[Xkaliber]

Homework Statement

$$\int_{-3}^{3}|t|e^{-jwt}dt$$

The Attempt at a Solution

I am not sure if I need to break this into two regions due to the abs value...

 

[CompuChip]

Yes, that is one approach:

$$\int_{-3}^3 |t| e^{-j \omega t} \, dt = \int_{0}^3 t e^{-j \omega t} \, dt - \int_{-3}^0 t e^{-j \omega t} \, dt$$
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

$$\int_{-3}^3 |t| \cos(\omega t) \, dt + j \int_{-3}^3 |t| \sin(\omega t) \, dt$$
and use (anti)-symmetries to reduce the problem before taking care of the absolute value.