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Math Help - Trigonometric Substitutions (fixed)

  1. #1
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    Trigonometric Substitutions (fixed)

    (a) If we use the substitution t = 6tan(θ), then which of the following integrals is equivalent to ?

    1















    (b) Use the integral in part (a) to evaluate


    Please show all steps and tank you!
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  2. #2
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    Sorry.
    Can't believe I did that.
    Last edited by chabmgph; February 2nd 2009 at 05:17 PM.
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  3. #3
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    Hello, choshiwara!

    (a) If we use the substitution t = 6\tan\theta,
    then which of the following integrals is equivalent to: . \int\frac{dt}{t^2\sqrt{t^2+36}}

    (1)\;\int\frac{\sec\theta\,d\theta}{36\tan^2\!\the  ta} \qquad (2)\;\int\frac{d\theta}{216\tan\theta\sec\theta} \qquad(3)\;\int\frac{d\theta}{36\tan^2\!\theta} \qquad(4)\;\int\frac{\sec^2\!\theta\,d\theta}{36\t  an^2\!\theta}

    Let: t \:=\:6\tan\theta \quad\Rightarrow\quad dt \:=\:6\sec^2\!\theta\,d\theta \quad\Rightarrow\quad \sqrt{t^2+36} \:=\:6\sec\theta


    Substitute: . \int\frac{\overbrace{6\sec^2\!\theta\,d\theta}^{dt  }}{\underbrace{36\tan^2\!\theta}_{t^2}\cdot\underb  race{6\sec\theta}_{\sqrt{t^2+36}}} \;\;=\;\; \int\frac{\sec\theta\,d\theta}{36\tan^2\!\theta} . . . answer choice (1)


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    We have: . \frac{1}{36}\int\frac{\sec\theta}{\tan^2\!\theta}\  ,d\theta \;=\;\frac{1}{36}\int\frac{\frac{1}{\cos\theta}}{\  frac{\sin^2\!\theta}{\cos^2\!\theta}}d\theta \;=\;\frac{1}{36}\int\frac{\cos\theta}{\sin^2\!\th  eta}\,d\theta

    . . = \;\frac{1}{36}\int\frac{1}{\sin\theta}\!\cdot\frac  {\cos\theta}{\sin\theta}\,d\theta \;=\;\frac{1}{36}\int\csc\theta\cot\theta\,d\theta \;=\;-\frac{1}{36}\csc\theta + C .[1]


    Back-substitute: . t \:= \:6\tan\theta \quad\Rightarrow\quad \tan\theta \:=\:\frac{t}{6} \:=\:\frac{opp}{adj}

    So \theta is in a right triangle with: . opp = t,\;adj = 6
    Using Pythagorus, we find that: . hyp = \sqrt{t^2+36}
    . . Hence: . \csc\theta \:=\:\frac{\sqrt{t^2+36}}{t}


    Therefore, [1] becomes: . -\frac{\sqrt{t^2+36}}{36t} + C

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