Improper integral or should i solve it?

Question

The problem is:

But as u can see, answers dont ask me to solve it, so i sure that, there`re another way to get the answer

Answer 1

Start by noticing that as $x\to\infty$,

$$\frac{\sqrt x+\sqrt[3]{\sin x}}{x^{5/3}+x\cos x}\sim x^{-7/6}$$

So it converges as $x\to\infty$ by p-series. It should then be obvious that it must be positive, since the integrand is positive over the values, so...