Let $f(x,t) = -x^3 e^{-tx^3}$
I'm trying to find a dominating, integrable function over $f$ for all $t \in \mathbb{R}^+$. Specifically, I'm looking for a function $h$ s.t. $\forall t > 0$, we have $|f(x,t)| \le h(x), \forall x \ge 0$. My original idea was to try to show that $min\{\frac{1}{t^2x^5}, 1\}$ was such a function, but it failed in light of this post: Exponential Function Question.
I have already shown that
$|-x^3 e^{-tx^3}| = |x^3||e^{-tx^3}| \le |-x^3||e^{-tx}|$
so that if I could find an integrable dominator of $|-x^3||e^{-tx}|$ my original objective would follow. Any ideas?