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Finding integrals of the product of trig functions

  1. Jun 21, 2011 #1
    1. The problem statement, all variables and given/known data
    I've come across integrals of exponential and trig functions and I have no idea how to do them. Integration by parts doesn't really work because they just derive into either e or another trig function.

    One of them is [itex]\int[/itex]sin(a)*sin(b - a)da
    Another is [itex]\int[/itex]e(a)*sin(a)da

    2. Relevant equations
    sin(x + y) =sinx(cosy) + siny(cosx)
    cos(x + y) = cosx(cosy) - sinx(siny)
    sin^2(x) = (1 - cos2x)/2

    3. The attempt at a solution
    I've tried to use the trig identity for sin(b-a), but that just gives an extremely long sin and cos statement that doesn't help. the one with e is even more confusing. How am I supposed to manually solve them?
    Last edited: Jun 21, 2011
  2. jcsd
  3. Jun 21, 2011 #2

    Char. Limit

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    For the second problem, let

    [tex]I = \int e^{a} sin(a) da[/tex]

    Then integrate the right side by parts twice, and note that you get, as your integral, I. From there, just solve for I.
  4. Jun 21, 2011 #3


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    Another trick for integrating integrals like ∫exsin(x)dx is to instead do the integral ∫exeixdx =∫e(1+i)xdx as a simple exponential. Rationalize the result and note your original integral is the imaginary part. This avoids two integrations by parts and gives you the ∫excos(x)dx from the real part as a free bonus.
  5. Jun 21, 2011 #4
    thanks for the replies, found the answers. i just didn't go far enough with the identities and integration by parts.
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