prove/disprove absolute convergent

Question

$\int_0^\infty 3^{-x}x^4cos(2x)dx$

I succeeded to prove that this integral is conditionally convergent with Dirichlet's test. I don't know how to prove/disprove absolutely convergent..

Thanks !

Answer 1

You have that $$\lim_{x\to \infty }x^23^{-x}x^4\cos(2x)=0,$$ and thus $$3^{-x}x^4\cos(2x)=\mathcal O\left(\frac{1}{x^2}\right),$$ at the neighborhood of $+\infty $. Therefore it's absolutely integrable on $[1,+\infty )$. The integrability on $[0,1]$ is obvious. The claim follow.