About the convergence and divergence of the following integrals.

Question

I have these two integral and wanted to know how I can explain their convergence and divergence by using comparison test.

Answer 1

First integral diverges, because the integrand goes as $1/(1-w)$ as $w \to 1^-$. This is a non-integrable singularity. In fact, at a pole $w_0$, if the integrand goes as $1/(w-w_0)^a$ as $w \to w_0$, then we have convergence only if $a < 1$.

Second integral converges because, at the pole $\alpha_2$, the integrand goes as $1/(\alpha_2-w)^{1/2}$ as $w \to \alpha_2^-$, unless $\alpha_1 \lt 0$ and $\alpha_2 \gt 0$, in which case it diverges.