# Thread: Fraction integrand with square root

1. ## 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!

2. 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 $\displaystyle x+\frac{1}{2} = \frac{\sqrt{3}}{2}\sinh{\varphi}$.