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Thread: Fraction integrand with square root

  1. #1
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    Fraction integrand with square root

    Hi, I am having some trouble with this integral:
    $\displaystyle \int \frac{x}{\sqrt{x^{2}+x+1}} dx$

    I suspect I have to make a substitution, but I am unsure about what to substitute. I guess whatever it is it have to be something that gets rid of the square root? Or is that unneccesary?
    Thanks!
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  2. #2
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    This integral appeared few days ago in the forum? Anyway, rewrite the integral as:

    $\displaystyle \displaystyle I = \int\frac{x}{\sqrt{x^2+x+1}}\;{dx} = \int\frac{\frac{1}{2}(2x+1)+x-\frac{1}{2}(2x+1)}{\sqrt{x^2+x+1}}\;{dx}$

    $\displaystyle \displaystyle \Rightarrow I = \int\frac{\frac{1}{2}(2x+1)}{\sqrt{x^2+x+1}}\;{dx} +\frac{1}{2}\int\frac{2x-(2x+1)}{\sqrt{x^2+x+1}}\;{dx}$

    $\displaystyle \displaystyle \Rightarrow I = \int\frac{\frac{1}{2}(2x+1)}{\sqrt{x^2+x+1}}\;{dx} }}-\frac{1}{2}\int\frac{1}{\sqrt{x^2+x+1}}\;{dx}} $.

    Let $\displaystyle u = \sqrt{x^2+x+1}$ for the first one, and for the other complete the
    square $\displaystyle x^2+x+1 = \left(x+\frac{1}{2}\right)^2+\frac{3}{4}$, then let $\displaystyle x+\frac{1}{2} = \frac{\sqrt{3}}{2}\sinh{\varphi}$.
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