divergence of $\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$ i ran into this question and im sitting on it for a long time.
why does this integral diverge:
$$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$$
thank you very much in advance.
yaron.

Notice the following:
$(x^2 - x - 2) = (x - 2)(x + 1)$
Therefore notice that:
$\frac{2}{x^2 - x - 2} = \frac{2}{(x-2)(x+1)} $
From here we can do a partial fraction decomposition which basically means we want to find:
$A, B$ such that $\frac{A}{x-2} + \frac{B}{x+1} = \frac{2}{x^2 - x - 2}$
This alternatively means that:
$A(x+1) + B(x-2) = 2$
And therefore:
$Ax + Bx = 0 ==> A + B = 0$
$A - 2B = 2$
We solve this system of 2 linear equations in 2 unknowns to find that:
$A = \frac{2}{3}, B = -\frac{2}{3}$
So finally we have:
$\frac{2}{x^2 - x - 2} = \frac{\frac{2}{3}}{x-2} - \frac{\frac{2}{3}}{x+1}  $
From here it is obvious that by integrating this expression you are integrating over the asymptote of x - 2 and therefore the sum diverges
Hope that made sense!
