Is $\int_1^{\infty}\frac{x \cos(x)^2}{1+x^3}$ convergent or divergent?

Question

For the integral $$I= \int_1^{\infty}\frac{x \cos^2(x)}{1+x^3},$$ how do I test this for convergence or divergence?

I know that this an improper integral- however it cannot be solved so would need to use a comparison test for this.

Would the comparison test consist of: If $\cos(x)<1$ then we can use $1/(1+x^3)$ to show that it converges?

However... How can i compare the equation where there is an $x$ on the numerator of the original equation? Would I need to use something else to compare it with instead? Thanks.

Answer 1

Consider comparing like this: $$\frac{x\cos^2 x}{1+x^3} \le \frac{x}{1+x^3} \le \frac{1}{x^2}$$

You can also do limit comparison to $\frac{x}{x^3}$.

Answer 2

Notice that $\cos^2(x) \leq 1$, therefore $$\frac{x \cos^2(x)}{1+x^3} \leq \frac{x}{1+x^3}.$$ You also know that $$\frac{1}{1+x^3} \leq \frac{1}{x^3}$$ for $x \in (1, \infty)$. Therefore $$\frac{x \cos^2(x)}{1+x^3} \leq \frac{x}{x^3} = \frac{1}{x^2}.$$