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Math Help - Evaluate this inverse trig integral PLZZ!!!?

  1. #1
    HMV
    HMV is offline
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    Evaluate this inverse trig integral PLZZ!!!?

    f(x)=1/(x(9x^2-4)^(1/2))
    Please show me your steps.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by HMV View Post
    f(x)=1/(x(9x^2-4)^(1/2))
    Please show me your steps.
    You have \int\frac{dx}{x\sqrt{9x^2-4}}

    Instead of making a trig sub try using a reciporcal substitution first. Let x=\frac{1}{\varphi}, so our integral becomes

    \begin{aligned}\int\frac{dx}{x\sqrt{9x^2-4}}&\stackrel{x=\frac{1}{\varphi}}{\longmapsto}\in  t\frac{\frac{-1}{\varphi^2}}{\frac{1}{\varphi}\sqrt{\frac{9}{\va  rphi^2}-4}}\\<br />
&=-\int\frac{d\varphi}{\sqrt{9-4\varphi^2}}\end{aligned}

    Now this is a much more common case of letting 2\varphi=3\sin(\phi)

    See how that goes.

    You could also have in the initial step let either 3x=2\sec(\phi) or 3x=2\cosh(\phi)
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