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Homework Help: Quotient rule integration problem

  1. May 11, 2010 #1
    [tex]\int \frac{1}{2x+3}=\frac{\ln |2x+3|}{2}+c[/tex]

    so why is [tex]\int \frac{1}{x^2+x}\neq \frac{\ln |x^2+x|}{2x+1}+c[/tex] ?

    is it because in general ,

    [tex]\int \frac{1}{x}=\ln |x|+c[/tex]

    the denominator is meant to be only linear function ?
     
  2. jcsd
  3. May 11, 2010 #2

    Mentallic

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    Homework Helper

    Re: integration

    Yes, because to go the other way, that is, take the derivative of the result [tex]\frac{ln|x^2+x|}{2x+1}[/tex] you need to use the quotient rule. It's not as simple as treating 2x+1 as a constant, which is what you instead get if the function in the log is linear.

    e.g.

    [tex]\int\frac{1}{ax}dx=\frac{ln|ax|}{a}[/tex]

    [tex]\frac{d}{dx}\left(\frac{ln|ax|}{a}\right)=\frac{1}{ax}\frac{a}{a}=\frac{1}{ax}[/tex]

    While

    [tex]\int\frac{1}{ax^2}dx \neq \frac{ln|ax^2|}{2ax}[/tex]

    [tex]\frac{d}{dx}\left(\frac{ln|ax^2|}{2ax}\right)=\frac{\frac{1}{ax^2}.2ax-2a.ln|ax^2|}{4a^2x^2} \neq \frac{1}{ax^2}[/tex] as required.
     
  4. May 11, 2010 #3
    Re: integration


    thanks , so it only works when the denominator is a linear function .
     
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