Question

Determine if the improper integral converges or diverges. Justify your answer by making a prediction, a correct inequality, and a conclusion.

Answer 1

\[\int\limits_{2}^{\infty}\frac{ x^{5} }{ x^{6}-1 }dx\]

Answer 2

I think you may be able to say that for large values of x, \[x^{6}+1 \approx x^{6}\]and then divide, but I am unsure if that would be permissible.

Answer 3

I think that logic works. Normally, you would just divide the highest powers of x in the numerator and denominator (in this case, x^5 and x^6) by the highest power of x in the numerator (x^5). Then apply the p-test.