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Math Help - trig substitution problem

  1. #1
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    trig substitution problem

    can someone solve

    ∫ square root of (9-4x^2dx)
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  2. #2
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    Quote Originally Posted by cummings15 View Post
    can someone solve

    ∫ square root of (9-4x^2dx)
    How about
    \int\sqrt{9-4x^2}\, dx=3\int \sqrt{1-\left(\tfrac{2}{3}x\right)^2}\, dx=\ldots

    Substituting  z := \tfrac{2}{3}x, thus dz = \tfrac{2}{3}dx and dx = \tfrac{3}{2}dz. This gives you

    =3\int \sqrt{1-z^2}\cdot \tfrac{3}{2}\,dz=\tfrac{9}{2}\int \sqrt{1-z^2}\, dz

    and, maybe, you kow how to solve this last integral...
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  3. #3
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    i think the final answer is

    (2/3)(9-4x^2)(3/2) + C
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  4. #4
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    Quote Originally Posted by cummings15 View Post
    i think the final answer is

    (2/3)(9-4x^2)(3/2) + C
    No, I don't think so, because you have

    \int \sqrt{1-z^2}\, dz=\frac{1}{2}\left(\arcsin(z)+z\sqrt{1-z^2}\right)+C

    which is perhaps a little more complicated than you thought.
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  5. #5
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    ok so i tried solving that and i got

    (9/4)arcsin(2x/3)+2x(square root of 9-4x^2)+c
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  6. #6
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    Quote Originally Posted by cummings15 View Post
    ok so i tried solving that and i got

    (9/4)arcsin(2x/3)+2x(square root of 9-4x^2)+c
    Why don't you just use Wolfram Mathematica Online Integrator to check your result? - If it is not quite correct, then at least it is close...
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  7. #7
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    here are my choices

    2/3(9-4x^2)^3/2 + C

    1/3tan^-1(2x/3)+c

    square root of (9-4x^2)^3/2 divided by 8x

    9/4arcsin(2x/3) + 2x square root of (9-4x^2) + C



    i think it is either the first or last choice

    it didn't show up in wolfram
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  8. #8
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    Quote Originally Posted by cummings15 View Post
    here are my choices

    2/3(9-4x^2)^3/2 + C

    1/3tan^-1(2x/3)+c

    square root of (9-4x^2)^3/2 divided by 8x

    9/4arcsin(2x/3) + 2x square root of (9-4x^2) + C



    i think it is either the first or last choice
    I get (as does Wolfram's Integrator): \int \sqrt{9-4x^2}\, dx=\tfrac{9}{4}\arcsin\left(\tfrac{2x}{3}\right)+{  \color{red}\tfrac{x}{2}}\sqrt{9-4x^2}+C

    it didn't show up in wolfram
    Well, you have to enter the integrand first: I could not get a link to the result because that particular link got mangled when I tried to insert it into my post.
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  9. #9
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    Quote Originally Posted by cummings15 View Post
    can someone solve

    ∫ square root of (9-4x^2dx)
    hi

    or make a substitution of x=3/2 sin theta
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