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Integral x^2(1-x^2)

  1. Feb 15, 2012 #1
    People, today i had a exam in math analysis and there was a integral to solve:

    ∫ x^2√(1-x^2) dx

    ok, i started to think about the trigonometric substitution. x= sint

    but, with that substitution now i have a ∫sin^2tcos^2t dt

    so i have to do something like ∫(1-cos^2t)(cos^2t) dt ok and i thought (no thanks...)
    i never learned how to solve a integral with the trigonometric formula, so solve something like
    ∫cos^4t dt takes a lot of time.

    So i tried t = √(1-x^2)

    dt/dx = -x/(√(1-x^2) )

    So now i have a integral

    -∫(x^2*t*√(1-x^2))/(x) dt
    -∫(x^2*t*t)/(x) dt
    -∫(x*t^2) dt

    as we know t = √(1-x^2) so x= 1-t^2

    -∫((1-t^2)t^2) dt = -∫t^2 - t^4 dt

    ok now it is easy...

    Please tell me that i did it in the correct way!
  2. jcsd
  3. Feb 15, 2012 #2


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    You can easily check by differentiating, however I think that you'll find you're off by a factor of x then.

    The problem seems to be
    if t2 = 1 - x2 then x2 = 1 - t2.
    I don't know how much that helps you though.
  4. Feb 15, 2012 #3

    what a stupid error.

    Damn. ok i should do it by trigonometric substitution.
  5. Feb 15, 2012 #4


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    I just made the observation that
    [tex]x^2 \sqrt{1 - x^2} \propto x \cdot \frac{d}{dx} (1 - x^2)^{3/2}[/tex]
    so maybe you can try partial integration.
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