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Math Help - Trig Substitution

  1. #1
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    Trig Substitution

    Evaluate: int. 1/[x sqrt.(9+x^2)] dx
    I am having trouble integrating it toward the end. Thanks for the help.
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  2. #2
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    Hello, nystudent2729!

    \int \frac{dx}{x\sqrt{9+x^2}}

    Let: x \,=\,3\tan\theta \quad\Rightarrow\quad dx \,=\,3\sec^2\!\theta\,d\theta


    Substitute: . \int\frac{3\sec^2\!\theta\,d\theta}{3\tan\theta\cd  ot3\sec\theta} \;=\;\frac{1}{3}\int\frac{\sec\theta}{\tan\theta}\  ,d\theta \;=\;\frac{1}{3}\int\frac{\frac{1}{\cos\theta}}{\f  rac{\sin\theta}{\cos\theta}}\,d\theta

    . . . . . . . = \;\frac{1}{3}\int\frac{d\theta}{\sin\theta} \;=\;\frac{1}{3}\int \csc\theta\,d\theta \qquad\hdots\;\text{Got it?}

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