$$ \int_o^\infty t^ae^{bt}dt $$ for a,b reals.
I guess I would have to separate this integral in many cases for different values of a and b.
I know that if b < 0,
$$ \int_o^\infty t^ae^{bt}dt > \int_o^\infty {t^a}dt $$, which diverges for any a. The other cases are more tricky. Any help would be appreciated.
If $b>0$ then $$\lim_{t \to +\infty} t^ae^{bt}=+\infty$$ for any $a$ and the integral diverges. If $b<0$ then the near $0$ you have $1<e^{bt}<2$ hence the integral converges if and only if $a>-1$, since there is no problem at infinity. For $b=0$ the integral never converges.