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Math problem integration by partial fractions

  1. Dec 7, 2015 #1
    1. The problem statement, all variables and given/known data
    integrate (4x+3)/(x^2+4x+5)^2

    2. Relevant equations


    3. The attempt at a solution
    I know to solve this problem you have to work with partial fractions, in the solution we were given they solve as followed

    4x+3=A(x^2+4x+5)'+B

    I don't know why they take the derivative of x^2+4x+5

    thank you in advance!
     
  2. jcsd
  3. Dec 7, 2015 #2

    Mark44

    Staff: Mentor

    From what you show here they are not using partial fractions. What they're doing is an ordinary substitution, with ##u = x^2 + 4x + 5##. In this substitution, what is du? That's where the derivative you're asking about comes in.
     
  4. Dec 7, 2015 #3

    epenguin

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    In order to solve the problem! I think I remember doing this myself for some recent problem on here.

    You have a linear divided by a quadratic. The derivative of a quadratic is a linear. If you are lucky, i.e. in the simplest case the numerator will be the derivative of the denominator (multiplied by a number). In general however the derivative of the quadratic will be the numerator (multiplied by a number) plus a constant. So you will get to integrate something of form

    C1Q'/ Q + C2/Q, where the C's are known constants.

    The integral of the first fraction is C1 ln Q , and the second fraction you expresse as partial fractions, and will get further ln 's .
     
  5. Dec 7, 2015 #4
    thank you very much for the explanation, I will try the exercise with your method
     
  6. Dec 7, 2015 #5

    epenguin

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    I expect someone else discovered it first. :oldsmile:
     
  7. Dec 7, 2015 #6

    SteamKing

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    To be sure, you have a linear polynomial divided by a quadratic polynomial squared. There might be a partial fraction expansion, but it will probably be messy.

    When the denominator is raised to a power, it's better to see if substitution can be used.
     
  8. Dec 8, 2015 #7

    epenguin

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    Oops yes :redface: I had missed that the denominator was squared. So it seems they want to get it in the form Q'/Q2 (multiplied by a number) which is -(1/Q)' .
    So that bit can be integrated to -1/Q and you are left with some number multiplying 1/Q2 which is a standard form to integrate.
     
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