Prove that $\int_{1}^{\infty} \cos(x)\cdot \frac{\sqrt{\ln(x)}\cdot x}{(x^2 + \sqrt[3]{\ln(x)})\cdot (x+5)}\,dx$ converges absolutely.

Question

I'm attempting to prove that $\int_{1}^{\infty} \cos(x)\cdot \frac{\sqrt{\ln(x)}\cdot x}{(x^2 + \sqrt[3]{\ln(x)})\cdot (x+5)}\,dx$ converges absolutely. I've tried using the comparison test a few times, but only ended up with diverging integrals.

Does anyone have an idea, as to how to approach this?

Thanks in advance!

Answer 1

$$\vert\cos(x)\cdot \frac{\sqrt{\ln(x)}\cdot x}{(x^2 + \sqrt[3]{\ln(x)})\cdot (x+5)}\vert\le\frac{\sqrt{\ln(x)}\cdot x}{x^2\cdot x}\le\frac{\sqrt{x-1}\cdot x}{x^3}\le \frac{x^{3/2}}{x^3}=x^{-3/2}$$

where the second inequality uses $\ln x \le x-1$ and the third uses $x-1<x$.

$x^{-3/2}$ is integrable on $x\ge1$ and we are done.