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Math Help - Trigonometric integration

  1. #1
    Junior Member
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    Trigonometric integration

    Hey, was wondering if anyone could help out with a pointer on how to do this in a more structured fashion.

    So the original task is to solve the indefinite integral of
    sqrt(1-x2)/x4

    So taking a cue from the square root i substitute
    x = sin(t)
    dx = cos(t)dt

    and get the integral of
    cos2(t)/sin4(t)

    And here my brain is going tilt, because I can't seem to figure out method will actually take me forward from there. What I did was working backwards, just playing around with trigonometric identities and finding that the derivative of
    -1/3tan3(t)

    gives the integrand. Back substituting, using that
    tan(t) = sqrt(1-x2)

    confirms that this is the correct answer.

    But yea, this isn't a very elegant way of doing it, so maybe someone could point out what I'm missing?
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  2. #2
    Senior Member Educated's Avatar
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    Re: Trigonometric integration

    It would help if you wrote out

    \dfrac{\cos ^2 (t)}{\sin^4 (t)} = \cot^2 (t) \csc^2 (t)

    So your integral will now be:

    \int \cot^2 (t) \csc^2 (t) dt

    If you let u = \cot (t)

    you will find that du = -\csc^2 (t) dt

    And then go from there...
    Last edited by Educated; April 10th 2013 at 05:08 AM. Reason: changed mistake cot^2 (t) to cot (t) and csc^2 (t) to -csc^2 (t)
    Thanks from Scurmicurv
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  3. #3
    Junior Member
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    Re: Trigonometric integration

    Aaaaah. The sad thing is that I think I started out by trying that but got the secondary trig identities mixed up. Anyhows, many thanks for helping out!
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