I have the following intergal: integral from 0 to infinity of (x^2)/(2x^3-x+1).
I do not know how to create an inequality that will help me determine this convergence. Also I have a general question: when i have something like (2x^n-x^sth+1) for example in the denominator, what is a general approach for bounding that? How do I determine what that is going to be smaller from when I have to work from x=0 to x=infinity? Thank you
Note that $0 < 2x^3 - x + 1 \leq 2x^3$ for all $x \geq 1$. Therefore $$\int_{1}^{\infty}\frac{x^2}{2x^3 - x + 1}dx \geq \int_{1}^{\infty}\frac{1}{2x}dx.$$ The integral on the right diverges, therefore so does the one on the left (and hence your integral diverges).