The question is to test whether $$\int^\infty_1\frac{x^{2012}-20x^7-14}{x^{2014}-30x^{11}+13}dx$$convergent or not. I have found that this rational function which is integrated from 2 to infinity is convergent, and I also founded that there is only one root $\alpha$ between 1 and 2 such that $\alpha^{2014}-30\alpha^{11}+13=0$.
After this, I use Wolfram Alpha to see this graph, and I was surprising that this improper integral is convergent.But I still have no idea how to test.
Now my question is I can not decide whether $$\int^\alpha_1\frac{x^{2012}-20x^7-14}{x^{2014}-30x^{11}+13}dx$$is convergent or not,
Thanks in advance for any help.
This is the situation of the integral near the root of $x^{2014}-30x^{11}+13$, Thanks to Cesareo.
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