Convergence/Divergence $\int_2^\infty \frac{4x^3+3x^2-x}{5x^5-2x^4+x^2-2}\ln x \, dx$

Question

$$\int_2^\infty \frac{4x^3+3x^2-x}{5x^5-2x^4+x^2-2}\ln x\, dx$$

How should I approach this? can I look at $$\int_2^\infty \frac{\ln x}{x^2}\,dx \text{?}$$

Answer 1

The integrand function is bounded between $\frac{4\log x}{5 x^2}$ and $\frac{84\log x}{65 x^2}$ on the interval $(2,+\infty)$, hence the integral is converging since $$ \int_{2}^{+\infty}\frac{\log x}{x^2}\,dx = \frac{1+\log 2}{2}.$$