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An integration involving inverse trig.

  1. Dec 29, 2009 #1
    1. The problem statement, all variables and given/known data

    The question is integrate

    [tex]\int_0^1 (2+\frac{1}{x+1}+\frac{2x+1}{x^2+4}) dx[/tex]

    2. Relevant equations

    [tex]\int_0^1 (2+\frac{1}{x+1}+\frac{2x+1}{x^2+4}) dx =2 + ln(\frac{5}{2}) +\frac{1}{2}arctan(\frac{1}{2})[/tex] (yes, it's a prove question)

    3. The attempt at a solution

    The first two parts I have no problem but I am not quite sure how to integrate the

    [tex]\frac{2x+1}{x^2+4}[/tex] part. I know it has to do with arctan but I'm not quite sure about the [tex]2x+1[/tex] part.

    Thanks in advance,
  2. jcsd
  3. Dec 29, 2009 #2
    Split up the last fraction so that you have [tex]\frac{2x}{x^{2}+4} + \frac{1}{x^{2}+4}[/tex]. Use a simple substitution for the first fraction and then the second fraction will involve the inverse tangent. In general,

    [tex]\int \frac{dx}{x^2+a^2} = \frac{1}{a} arctan(\frac{x}{a}) + C[/tex]
  4. Dec 29, 2009 #3
    ahh! I completely missed that step! Thanks for your help :)

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