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Inegration Frustration

  1. Jan 26, 2009 #1
    1. The problem statement, all variables and given/known data
    [itex]\int (9-x^2)^{5/2} dx [/itex]


    2. Relevant equations



    3. The attempt at a solution
    Letting[tex] x=3sin\theta[/tex] and [tex] dx=3 cos \theta[/tex],

    [itex] 243\int(1-sin^2\theta)^{5/2}3 cos \theta d\theta[/tex]

    Taking the 3 out gets

    [tex] 729 \int (1-sin^2 \theta)^{5/2} cos \theta d\theta [/tex]

    I'm not sure where to go from here. Do I combine everything algebraically and then do a u sub? Or do I integrate by parts here?
     
  2. jcsd
  3. Jan 26, 2009 #2

    AEM

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    Gold Member

    Well, you could remember that [tex] (1 - sin^2 \theta) = cos^2 \theta [/tex] so that your integrand becomes [tex] cos^6 \theta [/tex] and then look that integral up in a table of integrals, or look up a reduction formula for [tex] cos^6 \theta [/tex]. You'll be able to integrate the expression that results from the reduction formula.
     
  4. Jan 27, 2009 #3

    gabbagabbahey

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    Careful; [tex](\cos^2 \theta)^{5/2}=|\cos^5 \theta|\neq\cos^5 \theta[/tex] in general.
     
  5. Jan 27, 2009 #4

    AEM

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    Gold Member

    You are, of course, correct. I should have been more careful.
     
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