Calculus

Which of the following integrals cannot be integrated using partial fractions using linear factors with real coefficients?
a) integral of (x^2-1)/(x^3+x) dx
b) integral of 1/(9x^2-4) dx
c) integral of (x^3-x+3)/(x^2+x-2) dx
d) All of these can be integrated using partials fractions with linear factors and real coefficients

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  1. clearly they can all be handled using partial fractions.
    All of the denominators can be broken in linear and quadratic factors.
    (x^3+x) = x(x^2+1)
    (9x^2-4) = (3x-2)(3x+2)
    (x^2+x-2) = (x+2)(x-1)

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    posted by oobleck
  2. oops. I didn't see the part where it wanted only linear factors. So, the first one fails the test.

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    posted by oobleck
  3. It works if for the second fraction you use a linear expression, that is
    (x^2-1)/(x^3+x) = (x^2-1)/((x)(x^2+1) )
    let (x^2-1)/((x)(x^2+1) ) = A/x + (Bx+C)/(x^2 + 1)

    A(x^2 + 1) + x(Bx + C) = x^2 - 1
    let x = 0 ---> A + 0 = -1 or A = -1
    let x = 1 -----> 2A + B+C = 0
    B+C = 2 **
    let x = -1 ---> 2A - (-B+C) = 0
    B-C = 2 ***
    add ** and *** ----> 2B = 4
    B = 2 and C = 0

    so (x^2-1)/(x^3+x) = -1/x + (2x)/(x^2 + 1)
    ∫(x^2-1)/(x^3+x) dx
    = ∫ 2x/(x^2 + 1) dx - ∫ 1/x dx
    = ln(x^2 + 1) - lnx + c

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    posted by Reiny

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