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Simple substitution integral

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the following integral

    ∫1/(x*sqrt(x^2-1) dx


    2. Relevant equations



    3. The attempt at a solution

    I've decided to use the substitution:

    x = sec u
    dx = sec u * tan u du

    Substituting on the integral I got:

    ∫sec(u)*tan(u) / [sec u * sqrt((sec^2(u)-1))] du

    Since 1+tan^2(u) = sec^2(u) the integral simplifies to
    ∫ sec(u)*tan(u) / [sec(u)*tan(u)] du = ∫ du = u + c = sec(u) + c, c being an arbitrary constant.

    The answer on the solutions is given by the substitution

    u = sqrt(x^2-1)

    Is my answer wrong? Because it seems way simplier this way, and I don't see nothing wrong with the substitution...

    If anyone could help me I'd appreciate!

    Thanks.
     
  2. jcsd
  3. Jan 29, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    If the integral comes out to u+c, and you want to express it in terms of x, u=arcsec(x). u isn't equal to sec(u). And the answer on the solutions doesn't work. Test it by taking the derivative. arcsec(x)+c does work.
     
  4. Jan 29, 2013 #3
    I'm sorry but I'm a little confused. The result is obviously incorrect, but I don't think i did any wrong assumption as I was solving the integral. All the mathematical steps appear to be correct, including the substitution. How can it be wrong then?

    I solved it with the substitution from the solutions and I got to the result arctan(sqrt(x^2-1)), which is different from arcsin but is correct (I confirmed it with Matlab).
     
  5. Jan 29, 2013 #4

    Mark44

    Staff: Mentor

    Your work was fine up until you undid your substituion. If x = sec(u), then u = sec-1(x) or arcsec(x).

    When you undid your substitution, you replaced u with sec(u), which is incorrect. That's what Dick was saying.
     
  6. Jan 29, 2013 #5

    Dick

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    Science Advisor
    Homework Helper

    Your solution is not wrong. You got u+c. Since x=sec(u), u=arcsec(x). That is also correct. I read your post wrong. I thought you said the solution given was sqrt(x^2-1), but you didn't say that, you just said the subsitution was u=sqrt(x^2-1). Sorry!
     
  7. Jan 29, 2013 #6
    Ooh, I see it now... Thank you for your answer!
     
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