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Thread: why doesn't the denominator power get integrated with result ending with a ln ?

  1. #1
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    why doesn't the denominator power get integrated with result ending with a ln ?

    These two particular integrals are doing my head in ....

    $$\int_{}^{} (4x+4)/(2x^2+4x+5) dx$$

    which worked out to be
    ln(2x^2 + 4x + 5) + c

    and


    $$\int_{}^{} (x+1)/(2x^2+4x+5) dx$$

    which worked out to be

    1/ 4ln(2x^2 + 4x + 5) + c

    these where the answers in the worksheet answers ..I understand that when there's a fraction we need to us the natural log ...and then put the top x value over the bottom x value out the front of the natural log in the answer ,but the powers in the denominators have got me totally confused ....why don't we integrate that as well...?
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  2. #2
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    Re: why doesn't the denominator power get integrated with result ending with a ln ?

    You should know $\dfrac{d}{dx} \ln(u) = \dfrac{u'}{u}$, where $u$ is a function of $x$.

    the antiderivative of $\dfrac{u'}{u}$ is $\ln|u| + C$ ... that's it.

    For $\displaystyle \int \dfrac{x+1}{2x^2+4x+5} \, dx$, note the derivative of the denominator is $4x+4$. To get the numerator in that form, multiply the integrand by $\dfrac{4}{4}$ as follows ...

    $\displaystyle \dfrac{1}{4} \int \dfrac{4(x+1)}{2x^2+4x+5} \, dx = \dfrac{1}{4} \ln(2x^2+4x+5) + C = \ln\sqrt[4]{2x^2+4x+5} + C$

    note that absolute value is not necessary because $2x^2+4x+5 > 0$ for all $x$.
    Thanks from bee77
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    Re: why doesn't the denominator power get integrated with result ending with a ln ?

    I must admit I am a bit stumped by d/dxln(u) = u'/u is a function of x.
    the antiderivative of u'/u is ln|u| + C ....
    the way I think I did it was using the x value on the numerator and extracting the x value from the denominator with the integrated chain rule...and making a fraction of it....ie 4/4 first integral and 1/4 second one....they came out correct as the part in front of the ln....hmmm is that way ok as well or was it a total fluke I achieved that result...thanks
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    Re: why doesn't the denominator power get integrated with result ending with a ln ?

    $u=2x^2+4x+5$
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