Show that $I=\int_0^{\infty}e^{-a^2x^2}x^m\sin{nx}\,dx$ is absolutely converges for $m>0$

Question

Show that $$I=\int_0^{\infty}e^{-a^2x^2}x^m\sin{nx}\,dx$$ is absolutely converges for $m>0$.

$\int_0^{\infty}|e^{-a^2x^2}x^m\sin{nx}|\,dx\leq e^{-a^2x^2}x^m$

I want to use comparison test. How to check the convergence of $\int_0^\infty e^{-a^2x^2}x^m$.

Please help me to solve the remaining. Also suggest me whether the problem can be solved by Dirichlet's Test.

Answer 1

For $a\ne 0$, the $a$ can be set as $1$, or else, change of variable goes through.

For $e^{u}>u^{m+1}$ for large $u>0$, then $e^{-x^{2}}<x^{-2m-2}$ and we have $\displaystyle\int_{M}^{\infty}e^{-x^{2}}x^{m}dx\leq\int_{M}^{\infty}\dfrac{1}{x^{m+2}}dx<\infty$.