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Integrating sqrt(x^2+1)/x^2 using trig

  1. Mar 30, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]\int[/tex][tex]\frac{\sqrt{x^{2}+1}}{x^{2}} dx[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I tried a Trig Substitution with x=tan(u) and ended up with [tex]\int (csc(u))^{2} * sec(u) du[/tex]

    From here I am kind of stuck. I tried a few different integration by parts methods, but they got very messy. I also couldn't find any sort of table for this.
  2. jcsd
  3. Mar 30, 2008 #2


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    Hi dtl42! :smile:

    Yes, you're almost there …

    Put cosec^2 = 1 + cot^2, and you have:
    [tex]\int \left(sec(u)\,+\,cot(u)csc(u)\right) du\,.[/tex] :smile:

    (though personally, I'd have used x = sinhv, giving ∫coth^2(v)dv)
  4. Mar 30, 2008 #3
    Thanks very much, I can't believe I didn't see that. Its those variations of the Pythag. Identity that get me.
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