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Evaluate integral of a polynomial over another polynomial (complicated substitution)

  1. Dec 1, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex]\int^{√x}_{1}[/itex][itex]\frac{t^{3}+t-1}{t^{2}(t^{2}+1)}[/itex] dt

    2. Relevant equations

    3. The attempt at a solution

    So I first start by expanding the bottom part of the fraction to t[itex]^{4}[/itex]+t[itex]^{2}[/itex], and letting u equal to that. Then du=4t[itex]^{3}[/itex]+2t dt. I move the common multiple of 2 over to the other side so that it is (1/2)du=2t[itex]^{3}[/itex]+t dt. I cannot find out how to relate that to the numerator (although I am so close).

    Can someone please help? Thanks!
  2. jcsd
  3. Dec 1, 2011 #2


    Staff: Mentor

    Re: Evaluate integral of a polynomial over another polynomial (complicated substituti

    Do you know partial fractions decomposition? That seems to me to be the way to go. Using that technique you rewrite (t3 + t - 1)/(t2(t2 + 1) as the sum of three rational expressions of the form
    [tex]\frac{A}{t} + \frac{B}{t^2} + \frac{Ct + D}{t^2 + 1}[/tex]

    The idea is to find constants A, B, C, D so that the new representation is identically equal to the original rational expression. Once you find the constants, then integrate the sum of simpler functions.
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