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Math Help - Calc 1 integration substitution and Implicit diff!

  1. #1
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    Calc 1 integration substitution and Implicit diff!

    I need some help with 2 and 8.

    2) I see that U = sqrt(x-1) and then du=1/(2sqrt(x-1)) but how do I get an integer to sub in for xdx? Usually in our past problems we would have like xdx and u=x^2 so du = 2x and the fraction is 1/2. I guess I'm just not seeing it so some further good instruction would be nice.

    8.I have the answer key that my prof gave me and I can't get the answer. I keep coming up with:

    y=ln(x^2+y^2)
    y'=1/(x^2+y^2) * 2x +2yy'
    y'-2yy' = 2x/(x^2+y^2)

    but the answer key says:
    y'=2x/(x^2+y^2-2y)

    I don't see how you get the -2y in the denominator..
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  2. #2
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    \int x\sqrt{x-1} \, dx

    u = x-1

    x = u+1

    du = dx

    \int (u+1)\sqrt{u} \, du

    now integrate.


    y = \ln(x^2+y^2)

    y' = \frac{2x + 2yy'}{x^2 + y^2}

    x^2y' + y^2 y' = 2x + 2yy'

    x^2y' + y^2y' - 2yy' = 2x

    y'(x^2 + y^2 - 2y) = 2x

    y' = \frac{2x}{x^2 + y^2 - 2y}
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  3. #3
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    So when you integrate 2 you get:

    ((u+1)^2)/2 * (2/3U^(3/2))

    That seems really whacky and I'm guessing I'm not seeing where you're coming from. Another guy I asked said let U = sqrt(x-1) then solve the polynomial but I didn't understand that either. Sorry; I'm not this big of a dunce usually.
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  4. #4
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    Sorry to bump but I really need to know how to do number 2. I have no idea how to do that and I get an incredibly hairy equation when I take the gen. integral in my calc.
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  5. #5
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    \int_0^1 (u+1)\sqrt{u} \, du

    distribute the \sqrt{u} ...

    \int_0^1 u^{\frac{3}{2}} + u^{\frac{1}{2}} \, du

    now can you integrate?
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