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Double Substitution Correct or Faulty?

  1. Mar 7, 2014 #1
    Hey. I was reviewing some trig substitution and I know how to solve the problem but I was curious if you can also solve it by using two different substitutions/change of varaibles? If so, how would you relate the functions? Consider for example:

    ∫ (1-x^2)^1/2 dx, -1<= x <= 1

    x = sin Θ
    dx = cos Θ dΘ

    -pi/2 <= Θ <= pi/2
    ∫ (1-x^2)^1/2 dx = ∫ 1 - sin^2Θ dΘ

    u = 2Θ
    du = 2 dΘ

    -pi <= u <= pi
    Then the integral is equal to:

    Θ - 1/2Θ - 1/2 sin u(Θ(x)) + C

    where Θ(x) = sin^-1x.

    Would u(Θ(x)) = 2arcsinx be true?

    Which would give us:

    arcsin x - 1/2arcsin x - 1/2 sin (2arcsin x) + C

    However ugly it is, would this be correct with the adjusted interval? Does it make sense?
  2. jcsd
  3. Mar 7, 2014 #2


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    Hi Nano-Passion! :smile:
    sorry, not following this :confused:

    you've left out the √, and the cos Θ dΘ

    (and i think later on you're confusing 2Θ with 2Θ)
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