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.

$$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2} = \int_{2}^{\infty}\frac{dx}{(x - 2)(x+1)}$$
Hence at $x = 2$, the integrand is undefined (the denominator "zeroes out" at $x = 2, x = -1$. So $x = 2$ is not in the domain of the integrand. Although we could find the indefinite integral, e.g., using partial fractions, the indefinite integral of the term with denominator $(x - 2)$ would be $$A\ln|x - 2| + C$$ which is also undefined when $x = 2.$
Recall, the limits of integration are exactly those: limits.  $$\lim_{x\to 2} (A\ln|x - 2| + C) = -\infty$$ and hence, since the limit diverges, so too the integral diverges.
