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Calc II Integration and Completing the Square

  1. Sep 22, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]\int[/tex][tex]\frac{-\frac{1}{3}x+\frac{2}{3}}{x^{2}-x+1}[/tex] dx

    2. Relevant equations
    Completing the square, partial fractions

    3. The attempt at a solution
    I think I need to complete the square to do this but I can't figure out how to do it. Also, do I need to separate the numerator in doing this?

    This is the result of partial fractions so it is one of the last steps in my problem but I cannot understand how to do it. Any help would be fantastic!!
  2. jcsd
  3. Sep 22, 2008 #2


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    Homework Helper

    Complete the square on the denominator, then think about how you could factor the numerator to resemble a portion of the denominator.
    To get more than this you'll need to post some work.
  4. Sep 22, 2008 #3
    Hi statdad. Did you try it? Has an interesting result, huh? :wink:

  5. Sep 22, 2008 #4
    ok so I got the completing the square but how on earth can I continue? I just don't see it...

    [tex]\int[/tex][tex]\frac{x-2}{(x-\frac{1}{2})^{2}+\frac{3}{4}}[/tex] dx
  6. Sep 23, 2008 #5
    I think if you split it up using partial fractions, it would be better.
  7. Sep 23, 2008 #6


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    Staff Emeritus
    Science Advisor

    Not "partial fractions" because the denominator does not factor but physicsnoob93 may mean just
    [tex]\frac{x-2}{(x-\frac{1}{2})^2+ \frac{3}{4}}= \frac{x}{(x-\frac{1}{2})^2+ \frac{3}{4}}- \frac{2}{(x-\frac{1}{2})^2+ \frac{3}{4}}[/tex]
    The first integral requires a fairly simple substitution and the second an arctangent.
  8. Sep 23, 2008 #7
    Well, I see how splitting it up makes more sense than tackling it, but I don't know what to substitute u for to get rid of both the (x - (1/2) and x. And for the arctangent, how do I go about that? I do know how to set up a trig substitution with a radical, when I must draw a triangle and find sec^2, but I am unsure in this context.
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