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Substitution and Proving the Identity - HELP

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    y = x^2 √(1 − x^2)

    substitution x = sin Ө

    Prove that that is

    1/4 * sin^2 2Ө dӨ


    Got close to solving it once but then messed it up. Please help through steps as I'm getting stuck at one particular point.
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    (Original post by Azland)
    y = x^2 √(1 − x^2)

    substitution x = sin Ө

    Prove that that is

    1/4 * sin^2 2Ө dӨ


    Got close to solving it once but then messed it up. Please help through steps as I'm getting stuck at one particular point.
    Surely you can just divide by sinx?
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    (Original post by G0TA4s1T1NN3R)
    Surely you can just divide by sinx?
    that would make it Sin x * Cot^2 x right?

    Where do I go from there.
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    (Original post by Azland)
    that would make it Sin x * Cot x * Cosx = Cos^2 x right?
    Yeah, then you can use D'Jickin's identity to resolve it from there
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    (Original post by G0TA4s1T1NN3R)
    Yeah, then you can use D'Jickin's identity to resolve it from there
    My bad in the last one. I forgot to divide the Cos x you get from changing dx to dӨ so its actually Sin x * Cot^2 x

    Lost on what to do after that.
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    (Original post by Azland)
    My bad in the last one. I forgot to divide the Cos x you get from changing dx to dӨ so its actually Sin x * Cot^2 x

    Lost on what to do after that.
    Why are you using cot here?

     \displaystyle y = \int \left(x^2\sqrt{1-x^2}\right)dx

     \displaystyle x=sin\theta

    Differentiate the above expression to express, dx, in terms of ,  \displaystyle d\theta

    In your integral, remember to use the identity,  \displaystyle  cos^2x=1-sin^2x
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    x = sin Ө
    dx/dӨ = cos Ө so you can replace dx with cosӨ dӨ
    √1-x^2 = √1-sin^2Ө=√cos^2Ө = cos Ө

    = sin^2ӨcosӨcosӨ d
    =sin^2Өcos^2Ө dӨ

    2sin^2Өcos^2Ө = sin2Ө
    2sinӨcosӨ = sin2Ө

    so if you do sin^2Өcos^2Ө x 4 = 4sin^2Өcos^2Ө
    which is equal to 2sinӨcosӨ2sinӨcosӨ = (sin2Ө)^2
    as you multiply by 4 you then have to divide by 4
    therefore
    1/4 sin^2 2Ө

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    (Original post by Pin)
    x = sin Ө
    dx/dӨ = cos Ө so you can replace dx with cosӨ dӨ
    √1-x^2 = √1-sin^2Ө=√cos^2Ө = cos Ө

    = sin^2ӨcosӨcosӨ dӨ
    =sin^2Өcos^2Ө dӨ

    2sin^2Өcos^2Ө = sin2Ө

    so if you do sin^2Өcos^2Ө x 4 = 4sin^2Өcos^2Ө
    which is equal to (sin2Ө)^2
    as you multiply by 4 you then have to divide by 4
    therefore
    1/4 sin^2 2Ө

    I reached this point myself initially but then got stuck.

    2sin^2Өcos^2Ө = sin2Ө

    Lost you here. Isnt Sin2Ө = 2SinӨCosӨ
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    (Original post by Pin)
    x = sin Ө
    dx/dӨ = cos Ө so you can replace dx with cosӨ dӨ
    √1-x^2 = √1-sin^2Ө=√cos^2Ө = cos Ө

    = sin^2ӨcosӨcosӨ dӨ
    =sin^2Өcos^2Ө dӨ

    2sin^2Өcos^2Ө = sin2Ө

    so if you do sin^2Өcos^2Ө x 4 = 4sin^2Өcos^2Ө
    which is equal to (sin2Ө)^2
    as you multiply by 4 you then have to divide by 4
    therefore
    1/4 sin^2 2Ө

    (Original post by Azland)
    2sin^2Өcos^2Ө = sin2Ө

    Lost you here. Isnt Sin2Ө = 2SinӨCosӨ
     \displaystyle sin2\theta = 2sin\theta cos\theta 

\displaystyle sin^22\theta = 4sin^2\theta cos^2\theta

     \displaystyle \frac{sin^22\theta}{4} = sin^2\theta cos^2\theta
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    Got it, thanks. !
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    Yep, you're right. Wrote down the right working on my piece of paper :/
    glad you still got it.

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