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Trig Substitution

  1. Aug 31, 2010 #1
    1. The problem statement, all variables and given/known data
    ∫√(4-x^2)/x dx


    2. Relevant equations



    3. The attempt at a solution
    a^2=4 u^2=x^2 ⇒ u=asinθ
    a=2 u=x
    x=2sinθ sinθ=x/2 (Our professor uses a triangle method which I won't draw)
    2cosθ=√(4-x^2)
    dx=2cosθ dθ

    ∫√(4-x^2)/x dx=∫2cosθ/2sinθ dθ
    =∫cosθ/sinθ dθ
    u=sinθ
    du=cosθ dθ
    =∫1/u du
    =ln|u| + C
    =ln|sinθ| + C

    ⇒resubstitute x for θ
    =ln|x/2| + C

    Is this correct?
     
  2. jcsd
  3. Aug 31, 2010 #2

    Mark44

    Staff: Mentor

    In a quick scan I don't see anything obviously wrong. You can check your answer by differentiating what you ended with. Its derivative should be sqrt(4 - x^2)/x.
     
  4. Aug 31, 2010 #3
    I tried that, but the problem is I get 1/x. If I check with my calculator I get:
    2ln|[√(4-x^2)-2)/|x|] + √(4-x^2).
     
  5. Aug 31, 2010 #4

    Mark44

    Staff: Mentor

    There's a mistake above. You should get ∫2cosθ/2sinθ * 2cosθ dθ. You seem to have forgotten to substitute for dx.
     
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