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Help with integral

  1. Jan 27, 2013 #1
    This is not a homework problem, but a problem in the textbook that is not required. I am doing this to get a handle on the topic.

    I am evaluating a Fourier transform, without tables, and need to evaluate this integral:

    [tex]
    \int e^{-t} * sin(2 \pi f_c t) * e^{-j2 \pi ft} dt
    [/tex]

    I have tried two methods: 1) integration by parts, and 2) integration after expressing the sine function as a complex exponentials. I get stuck in both cases.

    The asterisks are there to assist with clarity of spacing. Thanks for any help you can provide,
    J
     
  2. jcsd
  3. Jan 27, 2013 #2

    mathman

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    Gold Member

    Use Euler formula to get exp(-t)*trig function. This is a standard integral (find in table).

    Trig function: sin(at), integral = a/(1 + a2)
    cos(at), integral = 1/(1 + a2)
    (a > 0 for both)
     
  4. Jan 27, 2013 #3
    Thanks, I did try Euler's formula but then worked the integral out manually (attempted to...)

    So this integral I would find in the table exclusively, and never try without a table? The way the drill is stated in the book (not a homework problem.), I wonder if I am to work it out without a table.

    Thanks,
    J
     
  5. Jan 28, 2013 #4

    mathman

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    You can integrate by parts twice to get an equation involving the original integral.

    I(exp(-t)cos(at)) = 1 + aI(exp(-t)sin(at)) = 1 - a2I(exp(-t)cos(at))

    Similarly for sin(at) integral.
     
  6. Jan 28, 2013 #5
    Thank you
     
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