Why is this integral bounded?

Hint: Prove that $x^\beta e^{-cx^2}=o\big(e^{-x}\big)$ as $x\to\infty$ for any $c>0$. Then: $$\int_\delta^\infty x^\beta e^{-cx^2}\,dx\leq\int_\delta^N x^\beta e^{-cx^2}\,dx+\int_N^\infty e^{-x}\,dx<\infty$$ for some $N$, so the integral converges.

Using the crude estimate $\log(u)<u$, for $u>0$, we see that for $\beta>0$, $c>0$

$$u^\beta e^{-cu^2}=e^{-cu^2+\beta\log(u)}<e^{-c(u-\beta/2c)^2}e^{\beta^2/4c}$$

Hence, we can assert that for any $\delta>0$

$$\int_\delta^\infty u^\beta e^{-cu^2}\,du< e^{\beta^2/4c} \int_{\delta-\beta/2c}^\infty e^{-cu^2}\,du$$

Can you finish now?