Convergence of $\int_{0}^{1} \frac{\ln^{2} x}{x^{2}+x-2} \ dx $

Question

How do you show that $\displaystyle \int_{0}^{1} \frac{\ln^{2}x}{x^2+x-2} \ dx $ converges?

The singularity at $x=1$ is not an issue since it is removable.

But what about at $x=0$?

Answer 1

You can show that $$\int_0^1 dx\ \ln^n x$$ converges by changing variables (it's a gamma function). Now, your denominator is benign in a neighborhood of $x=0$, and you can do a simple comparison to show that your integrand is smaller in magnitude than the integrand mentioned above.