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Math Help - Trig Substitution Strategy

  1. #1
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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

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

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

    \sqrt{x^{2} - a} then  a\sec\theta
    Last edited by Jason76; July 20th 2014 at 01:13 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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    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

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

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

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

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

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

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

    is correct.

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