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Math Help - Trig Substitution

  1. #1
    Junior Member ReneG's Avatar
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    Trig Substitution

    I'm trying to integrate \int \frac{\sqrt{9-x^2}}{x} \,\, dx

    I let x = 3\sin \theta \Rightarrow dx = 3\cos \theta \,\, d\theta

    made the substitution \int \frac{3\cos \theta\sqrt{9(1-\sin^2 \theta)}}{3\sin \theta} \,\, d\theta

    and simplified to 3\int \frac{\cos^2 \theta}{\sin \theta} \,\, d\theta which is where I'm stuck.

    I tried to simplify the trig functions into \cot \theta \cos \theta or \cos^2\theta \csc \theta but nothing helped
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  2. #2
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    Re: Trig Substitution

    You can simplify like this:

    \int\frac{1-sin^2(\theta)}{sin(\theta)}d\theta=\int[{csc(\theta)-sin(\theta)] d\theta
    Last edited by adkinsjr; July 2nd 2013 at 11:44 AM.
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  3. #3
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    Re: Trig Substitution

    Try a hyperbolic substitution x=3th(theta), it will work.
    Last edited by HK47; July 2nd 2013 at 12:55 PM.
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  4. #4
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    Re: Trig Substitution

    The substitution

    u^{2}=9-x^{2}

    works reasonably well.
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  5. #5
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    Re: Trig Substitution

    There are a number of ways to do it, but the way ReneG attempted it works fine as well, it was just the remaining trig that had to be worked out.
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  6. #6
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    Re: Trig Substitution

    To: atkinsjr
    My calculus 1 class did not involve integrating csc(x), that is why I offered the simpler hyperbolic substitution.
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