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Improper integral + Maclaurin series problem

  1. Aug 2, 2005 #1
    Can you please offer any hints or suggestions on how to do these two problems:

    1) Find the Maclaurin series of (x^2 + 1)/(3x^2 + 2x - 1).

    Should I perform long-division first? I can't seem to find any repeating pattern...

    2) Evaluate the integral sqrt(12-4x-x^2) from x=2 to x=6.

    I realize this is an improper integral since f(x)<0 for x>2 but I am bamboozled as to how to set up and evaluate it as a limit.

    Any help is appreciated in advance...
  2. jcsd
  3. Aug 2, 2005 #2
    Heh, okay I figured out the first one. I decomposed it using partial fractions to obtain a simpler geometric sum and from then on it was easy.

    Now as for the second, when I complete the square I notice that it is a circle centered @ (-2,0) with radius 4 but it asks me to find the area OUTSIDE the circle, namely from (x=2 to x-6)? Does this mean that the integral is divergent?
  4. Aug 2, 2005 #3


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    Homework Helper

    for 1)
    -factor the denominator
    -expand in partial fractions
    -expand new denominators in geometric series
    for 2) that is not an improper integral, but an imaginary one
    then if you like you can find the indefinite integral of that in terms of roots and logs.
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