I need help determining whether $$\int_{-\infty}^{\infty}\frac{x^2+x^{\frac{1}{3}}-1}{e^{2x}}dx$$ converges or diverges.
I know that it diverges since when you approach $-\infty$ the function will shoot up. My plan is to consider [$-\infty$,2] since the function will be greater than 0 as it has a root at around -1.4.
However I am having trouble bounding the numerator from below. I know I can make the denominator bigger by making it $e^{3x}$ since $e^{2x} \le e^{3x}$ but the numerator is what is troubling me.
Any suggestions would help.
Hint: Write the integrand as $(x^2+x^{\frac{1}{3}}-1)e^{-2x}$ and show that both factors are bigger than $1$ for sufficiently negative $x$.