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Thread: Integral by partial fractions?

  1. #1

    Integral by partial fractions?

    Integral from 0 to 1
    (x^3-4x-10)/(x^2-x-6) dx

    I know I can't do partial fractions right away so i need to divide it right?

  2. #2
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    Yes, anytime the power of the numerator is greater than the power of the denominator you should divide.

  3. #3
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    I am willing to work this out for you, because I think that partial fractions are dinosaurs in calculus. Any CAS and most advanced calculators will give you the answer.
    [tex]\L \frac{{x^3 - 4x - 10}}{{x^2 - x - 6}} = x + 1 + \frac{2}{{x + 2}} + \frac{1}{{x - 3}}[/tex].

    Why do we still ask students to do this? What a waste of time!
    “A professor is someone who talks in someone else’s sleep”
    W.H. Auden

  4. #4
    i got x+1 how do you get the rest?




    Quote Originally Posted by pka
    I am willing to work this out for you, because I think that partial fractions are dinosaurs in calculus. Any CAS and most advanced calculators will give you the answer.
    [tex]\L \frac{{x^3 - 4x - 10}}{{x^2 - x - 6}} = x + 1 + \frac{2}{{x + 2}} + \frac{1}{{x - 3}}[/tex].

    Why do we still ask students to do this? What a waste of time!

  5. #5
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    When you divide you get:

    [tex]\L\\\frac{3x-4}{x^{2}-x-6}+x+1[/tex]

    You need the partial expansion for the first part.

    Factor the denominator:

    [tex]\L\\\frac{3x-4}{(x-3)(x+2)}[/tex]

    [tex]\L\\\frac{A}{x-3}+\frac{B}{x+2}=3x-4[/tex]

    [tex]\H\\A(x+2)+B(x-3)=3x-4[/tex]

    [tex]\L\\Ax+2A+Bx-3B=3x-4[/tex]

    Equate coefficients:

    [tex]\L\\A+B=3[/tex]
    [tex]\L\\2A-3B=-4[/tex]

    A=1 and B=2

    [tex]\L\\\frac{1}{x-3}+\frac{2}{x+2}[/tex]

  6. #6
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    If you are lazy like me you can find the partial fraction without solving any equations. The trick is to expand the fraction around the singular points and adding up all the expansions in which you only include the singular terms. Liouville's theorem then guarantees that the resulting expression is the desired partial fraction decomposition

    We have:

    [tex]f(x)=\frac{x^3-4x-10}{x^2-x-6}=\frac{x^3-4x-10}{(x-3)(x+2)}[/tex]

    Expand around the point [tex]x=3[/tex]. Near [tex]x=3[/tex] we have:

    [tex]f(x)=\frac{A}{x-3}[/tex] + higher order terms.

    Multiply both sides by [tex]x-3[/tex] and take the limit [tex]x\rightarrow 3[/tex]. The limit is trivial and you find A = 1. The expansion around [tex]x=-2[/tex] is

    [tex]f(x)=\frac{B}{x+2}[/tex] + higher order terms.

    And B is found to be 2 by multiplying both sides by [tex]x+2[/tex] and taking the limit [tex]x\rightarrow -2[/tex] (which is again trivial).

    Finally we must expand around [tex]x=\infty[/tex]. This is only necessary if the degree of the numerator is equal or larger than that of the denominator. Regular terms in this expansion have negative powers of x, singular terms have positive powers of x. The result can be obtained using long division, just like Galactus did above. But now you don't need to consider the remainder. You find the leading terms of the expansion around [tex]x=\infty[/tex] are

    [tex]x+1[/tex]

    Adding up all three expansions reproduces [tex]f(x)[/tex]:

    [tex]f(x)=x+1+\frac{1}{x-3}+\frac{2}{x+2}[/tex]

  7. #7
    so i integrated it to be x^2/2 +x+2ln(x+2)+ln(x-3)

    but now i have to plug in 1 and zero and it wont work!

  8. #8
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    Quote Originally Posted by summergrl
    so i integrated it to be x^2/2 +x+2ln(x+2)+ln(x-3)

    but now i have to plug in 1 and zero and it wont work!
    You need to put absolute value signs around the arguments of the logarithms:

    [tex]\L\int\frac{dx}{x-a}=\log\left |x-a\right |[/tex]

    This follows from the fact that the derivative of [tex]\log(-x)[/tex] is [tex]\frac{1}{x}[/tex]. You only need to make sure that the integration path does not cross the singularity of the fraction. In this case the integration interval is from zero to 1 and the singularities are at minus 2 and at 3.

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