Calculus

Find the integral from 0 to 2 of xsqrt(5-sqrt4-x^2)).
The hint said to use substitution of u=sqrt(4-x^2), and that I needed one more substitution, but I don't know how to do it.

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  1. Sorry I forgot the dx after the integral.

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  2. ∫[0,2] x√(5-√(4-x^2)) dx
    Let
    u = 5-√(4-x^2)
    du = x/√(4-x^2) dx
    so, x dx = (5-u) du
    and you now have
    ∫[3,5] √u (5-u) du

    and I'm sure you can handle that, eh?

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  3. Wait, why is it the integral from 3-5 instead of 0-2?

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  4. because now we are using u, not x.
    If, after integration on u, you substitute back into terms of x, then you would use the limits [0,2].

    When x=0, u=3 and when x=2, u=5.

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