Integration of irrational function

  1. Here's the question that I got stuck:

    [inte]sqrt[x/(a-x)] dx ......................................(*)

    I tried to use the following substitution
    u=sqrt[x/(a-x)] and .........................................(1)
    dx = 2u(1-a)/(1+u2)2 du................(2)

    sub (1) and (2) into (*), after a few steps, I got

    (2-2a)[inte]du/(1+u2) - 2(1-a)[inte]du/(u2+1)2

    The answer derived from the first part, (2-2a)[inte]du/(1+u2), contains tan -1 but the model answer of this question is
    -[squ](ax-x2) + a/2sin-1[(2x+a)/a] + C
    For the second part, I let u = tan θ and got a strange expression.

    Is my approach correct and is the final answer obtained from the above method differs the model answer only by the constant of integration ? Or am I using a wrong substitution?
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    Check your work on your substitution.
     
  4. u=sqrt[x/(a-x)]

    dx = 2au/(1+u2)2 du

    thanks
     
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