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Tricky Integration with Trig Substitution

  1. Feb 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate.

    [itex]\int(4-y)\sqrt{4-y^{2}}dy[/itex]

    I have the solution using CAS software here:

    [itex]2y\sqrt{4-y^{2}}+8sin^{-1}\frac{y}{2}+\frac{1}{3}(4-y^{2})^{3/2}[/itex]


    but I need to do this by hand. I have researched the usual trig methods but am having some difficulty. Can someone please help me find the right identity?
     
  2. jcsd
  3. Feb 21, 2012 #2

    Dick

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    Substituting y=2*sin(t) looks like a good starting point.
     
  4. Feb 21, 2012 #3
    I don't think it goes anywhere though, I get:

    [itex]8cos(t)-4sin(t)cos(t)[/itex]

    when it comes time to bring the original variable back in I get a mess of cos(arcsin(y/2)) but maybe I am missing something
     
  5. Feb 21, 2012 #4

    Dick

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    If y=2*sin(t) then dy=2*cos(t)dt. I think you are forgetting that. And yes, it does take some work to integrate. If you make a try at it and show your steps I'm sure someone will try to help.
     
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