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Math Help - Trig Substitution

  1. #1
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    Trig Substitution

    The problem is

    Integrate sqrt(9x^2-25)/x^3 dx


    To solve for this would I use a right triangle like so

    Trig Substitution-yes.jpg
    I also think you might need to use a sec identity where the sqrt(x^2-a^2)=asec(theta) but I'm not sure how to set that up.
    Last edited by goku900; March 23rd 2013 at 06:48 AM.
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  2. #2
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    Re: Trig Substitution

    The question is not clear as to what is given and what do you want to find out.
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  3. #3
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    Re: Trig Substitution

    Sorry, forgot to say that needs to be integrated.
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  4. #4
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    Re: Trig Substitution

    Quote Originally Posted by goku900 View Post
    The problem is

    Integrate sqrt(9x^2-25)/x^3 dx


    To solve for this would I use a right triangle like so

    Click image for larger version. 

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    I also think you might need to use a sec identity where the sqrt(x^2-a^2)=asec(theta) but I'm not sure how to set that up.
    It might be easier to use a Hyperbolic Substitution rather than a Trigonometric. Recall that \displaystyle \cosh^2{(x)} - \sinh^2{(x)} = 1 \implies \cosh^2{(x)} - 1 = \sinh^2{(x)}. Also \displaystyle \frac{d}{dx} \left[ \cosh{(x)} \right] = \sinh{(x)}.

    So in this case, a substitution \displaystyle x = \frac{5}{3}\cosh{(x)} \implies dx = \frac{5}{3}\sinh{(x)}\,dx is appropriate.
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  5. #5
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    Re: Trig Substitution

    You should have 3x on the hypotenuse.

    So your substitution is \frac{5}{3x}=\sin{\theta} or \frac{1}{x}=\frac{3}{5}\sin{\theta} and then \cot{\theta}=\frac{\sqrt{9x^2-25}}{5} or 5\cot{\theta}=\sqrt{9x^2-25}.

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