When is $\int_1^\infty \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\,\mathrm{d}x$ finite?

Question

How do I solve the following problem?

For which of the following values of $a$ is the integral $$ \int_1^\infty \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\,\mathrm dx $$ finite? $$ a < 9/2, \quad a > 6, \quad a > 9/2, \quad a < 6. $$

Answer 1

I would say $a>6$. The numerator $$ x+\sqrt{x+\ln(x+2)}\sim x\qquad\mbox{ as }x\to +\infty $$ while the denominator $$ (x^a+\cos x)^{1/3}\sim x^{a/3}\qquad\mbox{ as }x\to +\infty\ . $$ Hence the ratio $$ \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\sim \frac{1}{x^{a/3-1}}\ . $$ To ensure convergence at infinity, we need to impose $$ a/3-1 >1\Rightarrow a>6\ . $$