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Integral (x^2 + 7x + 12)/(x + 4)

  1. Dec 31, 2011 #1
    1. The problem statement, all variables and given/known data

    Hello. I have a problem with a innocent-looking integral:

    [tex] \int\frac{x^2 + 7x + 12}{x + 4} dx [/tex]

    It doesn't look like i can use the law of sines, because the numerator is of higher order than the denominator. It doesn't look like the numerator is a multiple of the denominator or vice versa. It doesn't look like I can split the relatively complex numerator to an useful product.

    It looks like it relies on some high school level math for solving polynomials / fractions, but my math is rather rusty in some places and I can't seem to recall anything. I could look for the method if I knew its name. Sometimes it's best to ask a human being.

    2. Relevant equations

    It's from introductory examples, which means I'm not yet allowed to use:
    - Integration by substitution
    - Integration by parts


    3. The attempt at a solution

    [tex]\int\frac{x^2 + 7x + 12}{x + 4} dx = \int \frac{x^2}{x + 4} + \frac{7x}{x + 4} + \frac{12}{x + 4} dx =[/tex]
    [tex]=\int \frac{x^2}{x + 4} + \frac{7x}{x + 4}dx + 12 \int \frac{dx}{x + 4} =[/tex]
    [tex]= \int \frac{x^2}{x + 4} + \frac{7x}{x + 4}dx + 12 \ln |x + 4| + C_1[/tex]

    No, I can't get very far.
     
  2. jcsd
  3. Dec 31, 2011 #2
    The top can be factored, so that the denominator is canceled.
     
  4. Dec 31, 2011 #3
    Oh, I gave up too soon.
    It's not too hard if you start with the assumption that (x + 4) is a factor in numerator. Case closed :-)
     
  5. Dec 31, 2011 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Even if (x+4) were not a factor of the numerator, you can still do it by dividing the numerator by the denominator giving you a quotient and remainder.
     
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