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Finding factors in order to use U sub

  1. Apr 21, 2014 #1
    1. The problem statement, all variables and given/known data


    [tex]\displaystyle{\int}\dfrac{x}{\sqrt{x^2+x+1}}~dx =

    \displaystyle{\int}\dfrac{(x+\frac 1 2)-\frac 1 2}{\sqrt{(x+\frac 1 2)^2+(\frac {\sqrt 3} 2)^2}}~dx[/tex]

    2. Relevant equations

    Given.

    3. The attempt at a solution

    The solution is given, but I'm not sure how it was found. Is there a method for finding those factors or is it trial and error/intuition?
     
  2. jcsd
  3. Apr 21, 2014 #2

    Mark44

    Staff: Mentor

    There is a method. They are completing the square in the quadratic in the denominator.
    x2 + x + 1 = x2 + x + (1/4) + (1 - 1/4)
    = (x + 1/2)2 + 3/4
    = (x + 1/2)2 + (√(3)/2)2

    Then they are working with the numerator to get it as you see it in the expression on the right of what you posted.
     
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