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Homework Help: Integration of a fraction.

  1. Oct 13, 2007 #1
    1. The problem statement, all variables and given/known data

    [tex]\displaystyle \int{\frac{dx}{a^2+\left(x-\frac{1}{x} \right)^2}} [/tex]

    2. Relevant equations


    3. The attempt at a solution

    This one looks a bit odd. Had the denominator been a^2 + x^2, it is in one of the standard forms, whose integral is [tex]\frac{1}{a} \atan{\frac{x}{a}} [/tex]. But the denominator is in the form of a^2 + u^2 (where u is a function of x). I did try some manipulations, but to no avail. I tried putting x as sin(theta), but got something like cos(theta)d(theta)/(a^2+cos^4(theta)/sin^2(theta)), which seems even more complex. If someone can just point me into the direction to look, I'll attempt the solution.

    Thank you,
  2. jcsd
  3. Oct 13, 2007 #2

    Gib Z

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    Homework Helper

    Expand the brackets, simplify, multiply the entire integral by x^2/x^2, factor the denominator and partial fractions.
  4. Oct 13, 2007 #3
    Thanks for the quick reply, I'm currently here,

    [tex]\displaystyle \int{\frac{x^2dx}{x^2(a^2-2)+x^4+1} [/tex]

    I don't see how I can factorize/simplify the denominator or the expression...?

    Last edited: Oct 13, 2007
  5. Oct 13, 2007 #4

    Gib Z

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    Homework Helper

    Well let [itex]a^2-2 =b[/itex] and [itex]u=x^2[/itex]. Now it resembles a nice quadratic equation =]
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