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Absolute Value Integration

  1. Apr 13, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]
    \int_{-3}^{3}|t|e^{-jwt}dt
    [/tex]


    3. The attempt at a solution

    I am not sure if I need to break this into two regions due to the abs value...
     
  2. jcsd
  3. Apr 14, 2009 #2

    CompuChip

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    Homework Helper

    Yes, that is one approach:

    [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.
     
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