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Evasive integral substitution

  1. Oct 29, 2008 #1
    I am staring at an integral of the form

    \int \frac{sin(at)}{(1 + bsin^{2}(at))^{1/2}} dt

    which I have generated for myself (in attempting to model the behaviour of a particle in an oscillating field). I can't see a sensible substitution to try, at present. I could hunt down a standard integral, perhaps, but I suspect something obvious is evading me...

    Any hints? Also, any suggestions for brushing up on solving integrals of this sort? I'm a bit rusty :-)

  2. jcsd
  3. Oct 29, 2008 #2


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    It's sin that makes this complicated, isn't it? That seems the obvious place to start.
  4. Oct 29, 2008 #3


    Staff: Mentor

    This might be useful: Change 1 + b*sin^2(at) into 1 + b*(1 - cos^2(at) = 1 + b - b*cos^2(at).

    You could then use the substitution u = cos(at), du = -a*sin(at)dt.

    Then your integral would be roughly du/(A - bu^2)^(1/2), and you might be able to find that in a table of integrals or, failing that, apply a trig substitution.

    Anyway, that's the direction I would go as a start.
  5. Oct 30, 2008 #4
    A good suggestion. Thank you. It will end up with an arc sin of a cos, I think, but perhaps that can be rewritten more elegantly...
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