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Use of substitution for integration

  1. Dec 8, 2014 #1
    I was wondering if there is a convenient way of checking if a substitution is correct or not.
    For example, I tried solving for ∫(1/(a^2-x^2)dx using two different substitutions, x=acosu and x=asinu giving different solutions. I got the integral as arcsin(x/a) using x=asinu and -arccos(x/a) using x=acosu. But which one of these is the correct one and how do I tackle this sort of problem if it arises again?
     
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  3. Dec 8, 2014 #2

    dextercioby

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    Differentiating back, of course. In your question, the 2 solutions are related through a constant, as they should be. Can you guess it? How about derive it?
     
  4. Dec 8, 2014 #3

    Stephen Tashi

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    As dextercioby indicates, antiderivatives of a function are not unique. if the two answers differ by a constant then they could both be correct antiderivatives. (Consider things like [itex] sin(\theta) = cos( \frac{\pi}{2} - \theta) [/itex].)
     
  5. Dec 8, 2014 #4

    mathman

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    arccos(x) + arcsin(x) = π/2.
     
  6. Dec 8, 2014 #5
    Alright, thanks for the input. I'm guessing that since sin(x)=cos(π/2-x), then:
    sin(x)=u, cos(π/2-x)=u
    arcsin(u)=x, arccos(u)=π/2-x
    So arcsin(u)+arccos(u)=x+π/2-x=π/2
    Since they differ by a constant ,then
    arcsin(x/a)+c=-arccos(x/a)
    arcsin(x/a)+arccos(x/a)=c=π/2
     
    Last edited: Dec 8, 2014
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