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March 30, 2017

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1. Evaluate the following integrals
(a) 􏰀 4x2 +6x−12 / x3 − 4x dx

  • Calculus - ,

    I will assume you meant
    (4x^2 +6x−12)/(x^3 − 4x)
    = (4x^2 + 6x - 12)/( x(x+2)(x-2) )

    split it into partial fractions

    let
    A/x + B/(x+2) + C/(x-2) = (4x^2 + 6x - 12)/( x(x+2)(x-2) )

    multiply by x(x+2)x-2)

    A(x^2-4) + Bx(x-2) + Cx(x+2) = 4x^2 + 6x - 12

    This must be true for all x's
    let x = 0
    -4A = -12
    A=3
    let x = 2
    8C = 16 + 12-12 = 16
    C = 2
    let x = -2
    8B = 16 + 12-12 = -8
    B = -1

    so ∫ (4x^2 + 6x - 12)/( x(x+2)(x-2) ) dx
    = ∫3/x dx + ∫-1/(x+2)dx + ∫2/(x-2) dx
    = 3ln x - ln(x+2) + 2ln(x-2) + k, where k is a constant

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