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Definite Integral

  1. Apr 27, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]\int_0^a {x\sqrt{x^2+a^2}\,dx}[/tex]

    Also, (A>0)

    3. The attempt at a solution

    Firstly, I set


    Then take the derivative,

    du=2x dx

    [tex]1/2\int_0^a {\sqrt{u}\,du}[/tex]

    Now I integrated. So

    (1/3) * [(x^2+a^2)^3/2] from a to 0.

    I ended up with


    This is where I get lost. It must have something to do with the (A>0). The answer in the book is:


    I can't see how to eliminate the a's to get the (2*sqrt(2)-1).

    Thanks for the help.
  2. jcsd
  3. Apr 27, 2008 #2
    [tex]\frac 1 3[(a^2+a^2)^\frac{3}{2}-a^3][/tex]

    [tex]\frac 1 3[(2a^2)^\frac{3}{2}-a^3][/tex]

    [tex]\frac 1 3[2^\frac{3}{2}a^2^\frac{3}{2}-a^3)][/tex]

    Continue simplifying.
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