How to integrate :Integral of $\int 1/(x^2+x+1)$ with substitution?

Question

How can I evaluate this integral with substitution :

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

Answer 1

write $$x^2+x+1$$ as $$\left(x+\frac{1}{2}\right)^2+\frac{3}{4}$$ and Substitute $$t=x+\frac{1}{2}$$ it is $$\frac{3}{4}\left(\left(\frac{2t}{\sqrt{3}}\right)^2+1\right)$$ and then Substitute $$y=\frac{2t}{\sqrt{3}}$$ this here should be a possible result: $$2/3\,\sqrt {3}\arctan \left( 1/3\, \left( 2\,x+1 \right) \sqrt {3} \right) +C$$

Answer 2

HINT.-Putting $\dfrac{2}{\sqrt3}(x+\dfrac12)=t$ you get your integral equal to $$\dfrac{2}{\sqrt3}\arctan\left(\frac{2x+1}{\sqrt3}\right)+Constant$$ because of $\int\dfrac{dt}{t^2+1}=\arctan t+C$