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Integration by Parts with Complex Exponentials

  1. May 22, 2013 #1
    I'm looking at this problem here. (Exam practice, move to homework if you want...)
    nZ4jRd1.png

    First part is easy, but it's the second part that I can't quite figure out.


    I'm trying to get from ∫xe2x cos(2x)dx to this answer:
    ¼e2x(x cos(2x) + x sin(2x) - ½sin(2x))+C
     
  2. jcsd
  3. May 22, 2013 #2

    pwsnafu

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    Do you know Euler's formula?
     
  4. May 22, 2013 #3

    HallsofIvy

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    [tex]cos(2x)= \frac{e^{2ix}+ e^{-2ix}}{2}[/tex]
    so
    [tex]\int xe^{2x} cos(2x)dx= \frac{1}{2}\int x(e^{2(1+ i)x}+ e^{2(1- i)x})dx[/tex]
     
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