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Please give any hints about this integral

  1. Jan 31, 2009 #1
    Hi guys, I and my colleague have been struggling with this integral for over 2 hours. :D

    Any hints about solving it are welcome.

    [tex]\int \frac{\sqrt{x^2 - 4}}{x^4}[/tex]


    Thanks in advance :)
     
  2. jcsd
  3. Jan 31, 2009 #2

    Dick

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    Looks like a trig substitution to me. Try u=2*sec(x).
     
  4. Feb 1, 2009 #3
    Hi Dick,

    thanks for your reply.

    Unfortunately, in some countries(like mine) the secant and cosecant trig functions are not taught(not even in mathematical schools).

    Aren't there any alternative approaches? Isn't there a way to use sin/cos/tg/cotg substitutions?

    Thanks!
     
  5. Feb 1, 2009 #4
    You could try [tex]u=2*sin^{-1}(x)[/tex]
     
  6. Feb 1, 2009 #5

    arildno

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    The simplest is to use the hyperbolic cosine substitution, x=2Cosh(u).

    Using this, you'll readily find a proper anti-derivative, namely:
    [tex]A(x)=\frac{1}{12}(\frac{\sqrt{x^{2}-4}}{x})^{3}[/tex]
     
  7. Feb 1, 2009 #6

    Mark44

    Staff: Mentor

    I find it difficult to believe that these functions aren't presented in some countries. If that's the case, though, here is how they're defined:
    secant(x) = sec(x) = 1/cos(x)
    cosecant(x) = csc(x) = 1/sin(x)
     
  8. Feb 1, 2009 #7

    Mark44

    Staff: Mentor

    Dagda, did you mean this as u = 2/sin(x)? That's different from 2*sin^(-1)(x), which is the same as 2*arcsin(x).
     
  9. Feb 2, 2009 #8
    I think my brain exploded and I accidentally read the wrong trig sub off the table, whatever happened you are of course correct. :smile:
     
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