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Integral problem: 1/((x^2)(sqrt(4-x^2)))

  1. Oct 20, 2013 #1
    1. The problem statement, all variables and given/known data

    ∫ dx/ x2 (√4-x2)

    2. Relevant equations
    x=2sinθ
    dx=2cosθdx
    sin2 θ= (1 + cos (2θ))/2

    3. The attempt at a solution
    First attempt:
    Plug in the trig sub and get:

    ∫2cosθdθ/4sin2θ2cosθ
    The two cos θ cancel leaving

    ∫ dθ/ (4sin2θ)

    Now I replace the sin2θ with (1-cos2θ)/2
    and get

    1/2 ∫dθ/(1-cos2θ)

    I use u sub with the cos2θ
    u=2θ
    du/2=dθ

    ∫cosudu= sin2θ/2

    I plug it back into the original equation

    1/2 [ 2/θ-sin2θ]= 1/(θ-2sinθcosθ)

    I retrieve the x from the original equation to get

    4/ arcsin(x/2) -(x)(√(4-x2))

    I may be doing something completely wrong, but I can't figure out where. Any hint on where to find the mistake? I keep practicing trig sub problems and I can not get any right. It might have to do with the derivation of the equations, but I am not sure.
     
    Last edited: Oct 20, 2013
  2. jcsd
  3. Oct 20, 2013 #2

    Dick

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    When you say "I use u sub with the cos2θ" things start going badly wrong. I'm not even sure what you are doing after that. I would go back to when you have 1/(4sin^2(θ)). That's the same as csc^2(θ)/4. There's an easy integral for csc^2(θ).
     
  4. Oct 20, 2013 #3
    Well I do not feel very smart.

    ∫csc2xdx/ 4= -cot x/4 +c

    go back to the triangle and it is -(√4-x2/ 4x) +c

    Thank-you so much!
     
  5. Oct 20, 2013 #4

    Dick

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    You're welcome and well done. You started ok. You just missed where to get off the substitution train.
     
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