Q:

Integrals of $\int^\infty _2 \frac{x}{x^3 -2\sin x}dx$

How do I calculate the convergence of $\int^\infty_2\frac{x}{x^3 -2\sin x}dx$ ?
I know in generally, integration of (n-dimentional polynomial)/(n+2-dimentional polynomial) will be converge. 

A:

$$\left|\int^2_\infty \frac{x}{x^3 -2\sin x}dx\right| \leq \int_2^\infty \left|\frac{x}{x^3 -2\sin x}\right|dx \leq \int_2^\infty \frac{x}{|x^3 -2\sin x|}dx=\int_2^\infty \frac{x}{x^3 -2x}dx=\dfrac{1}{2\sqrt{2}}\ln\dfrac12(2+\sqrt{2})^2$$
Note:
$$|x^3 -2\sin x|\geq|x^3| -2|\sin x|\geq|x^3| -2|x|=x^3-2x$$

