How can I solve rational integrals if I cannot factor the denominator?

Question

Here is this integral and I dont know how do I start it I tried factoring the denominator but its not possible what should I do ?

$\int \frac{1}{x^2+x+1} dx$

Answer 1

In cases like this you can complete the square

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

$I = \displaystyle \int\dfrac{1}{(x+\frac12)^2+\frac34}\,dx$

$I = \frac{2}{\sqrt 3}\arctan\bigg(\frac{x+\frac12}{\frac{\sqrt3}2}\bigg)+C$

Answer 2

This is easy to solve after you've noticed that$$\frac1{x^2+x+1}=\frac1{\left(x+\frac12\right)^2+\frac34}.$$