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Math Help - use trig substitution

  1. #1
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    use trig substitution

    show that integral 1/x^2squareroot(x^2-a^2) =
    ((squareroot(x^2-a^2))/a^2x)+C using a trigonometric substitution
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  2. #2
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    Quote Originally Posted by mikegar813 View Post
    show that integral 1/x^2squareroot(x^2-a^2) =
    ((squareroot(x^2-a^2))/a^2x)+C using a trigonometric substitution
    I remember that sin^2(\theta)+ cos^2(\theta)= 1 so that, dividing by cos^2(\theta), tan^2(\theta)- 1= sec^2(\theta) and then a^2tan^2(\theta)- a^2= a^2sec^(\theta).

    That suggests using the substitution x= a tan(\theta) to get a "perfect square" inside the square root.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    I remember that sin^2(\theta)+ cos^2(\theta)= 1 so that, dividing by cos^2(\theta), tan^2(\theta)- 1= sec^2(\theta) and then a^2tan^2(\theta)- a^2= a^2sec^(\theta).

    That suggests using the substitution x= a tan(\theta) to get a "perfect square" inside the square root.
    Typo... HoI surely means

    dividing by cos^2(\theta), tan^2(\theta) + 1= sec^2(\theta) and then a^2 \sec^2(\theta) - a^2 = a^2 \tan^2(\theta).

    That suggests using the substitution x= a \sec(\theta) to get a "perfect square" inside the square root


    Edit:

    Just in case a picture helps...



    ... where



    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    Carry on, anti-clockwise... g(theta) should be a nice simplification with a straightforward integral G. Then use a triangle or whatever to map back to x. Actually, as F(a sec theta) = G(theta) you can map from G...





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    Last edited by tom@ballooncalculus; October 13th 2009 at 07:39 AM.
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