Integration of rational function ($ \frac{2x + 7} {x^2 + x + 1} $)

Question

I've attempted to integrate the function: $$ \frac{2x + 7} {x^2 + x + 1} $$ I've tried several techniques, but none of them are working! I want to try trig substitution, but I don't know what I can substitude.

Answer 1

Complete the square in the denominator:

$$x^2 + x + 1 = x^2 + x + \frac 14 + \frac 34 = \left(x+ \frac 12\right)^2 + \left(\frac{\sqrt 3}2\right)^2$$

Put $$x + \frac 12 = \frac{\sqrt 3}2\tan \theta$$

Answer 2

Hint

$$\frac{2x + 7} {x^2 + x + 1}=\frac{2x + 1+6} {x^2 + x + 1}=\frac{2x + 1} {x^2 + x + 1}+\frac{6} {x^2 + x + 1}$$

You recognize the integral of the first term. For the second, complete the square.