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Integration with partial fractions

  1. Jul 22, 2006 #1
    stuck on this one question. mostly cause I don't know the proper steps for using partial fractions.

    (4x-4)/(x^4 -2x^3 +4x^2 -6x +3)
    which factors to
    now I have the answer here. but I don't know the rules for decomposing this fraction. can someone go over them for me
    this decomposes to A + B/(x-1)^2 + Cx+D/(x^2+3)
    i don't understand this step, why is the A value by itself, and what is the determining factor that you have 2 terms in the numerator. I though it had something to do with the degree of the denominator. but then why is it B/(x-1)^2 for the one but Cx+D/(x^2+3) for the other. both are x^2 terms.

    hope i've explained myself well enough,
  2. jcsd
  3. Jul 22, 2006 #2


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    Well, first simplify your expression as follows:

    Now, assume an expansion of the form:
  4. Jul 22, 2006 #3


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    The general "rules" (which I am sure are in your textbook) are these:
    If a factor in the denominator is linear, i.e. (x-a), then you will need a fraction of the form [itex]\frac{A}{x-a}[/itex]. That is, the numerator is a constant because the denominator is first degree.
    If a factor in the denominator is a power of a linear term, i.e. (x-a)n, then you will need several fractions of the form [itex]\frac{A}{x-a}[/itex], [itex]\frac{B}{(x-a)^2}[/itex], up to the nth power: [itex]\fra{Z}{(x-a)^n}[/itex]. Again, the denominator is always a constant because the "base" factor, x-a, is first power.
    If a factor is quadratic and can't be factored, say ax2+ bx+ c, then you will need a fraction of the form [itex]\frac{Ax+ B}{ax^2+ bx+ c}[/itex]. In each case, the numerator may be degree one less than the denominator (of course, A might be 0). If it were not, then you could divide it out.
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