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Thread: Trig Substitution Strategy

  1. #1
    MHF Contributor Jason76's Avatar
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    Trig Substitution Strategy

    When the variable under the square root is not a square, then what is the general strategy? Any example?

    If the variable is a square then

    $\displaystyle \sqrt{a - x^{2}}$ then $\displaystyle a\sin\theta$

    $\displaystyle \sqrt{a + x^{2}}$ then $\displaystyle a\tan\theta$

    $\displaystyle \sqrt{x^{2} - a}$ then $\displaystyle a\sec\theta$
    Last edited by Jason76; Jul 20th 2014 at 12:13 AM.
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  2. #2
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    Re: Trig Substitution Strategy

    If it's a linear function under the square root, just substitute u for that entire linear function.
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    MHF Contributor Matt Westwood's Avatar
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    Re: Trig Substitution Strategy

    Quote Originally Posted by Jason76 View Post
    When the variable under the square root is not a square, then what is the general strategy? Any example?

    If the variable is a square then

    $\displaystyle \sqrt{a - x^{2}}$ then $\displaystyle a\sin\theta$

    $\displaystyle \sqrt{a + x^{2}}$ then $\displaystyle a\tan\theta$

    $\displaystyle \sqrt{x^{2} - a}$ then $\displaystyle a\sec\theta$
    Wrong, I'm afraid:

    $\displaystyle \sqrt{a^2 - x^{2}}$ then $\displaystyle a\sin\theta$

    $\displaystyle \sqrt{a^2 + x^{2}}$ then $\displaystyle a\tan\theta$

    $\displaystyle \sqrt{x^{2} - a^2}$ then $\displaystyle a\sec\theta$

    is correct.

    One good reason for using $\displaystyle a^2$ rather than $\displaystyle a$ is because then it's obvious what the sign is. $\displaystyle a^2$ is always positive.

    You should be able to find all this stuff in a basic primer on calculus: there is invariably a list of "useful" substitutions if the book you're using is worth a damn. If not, then find them on line.
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