For example: $$\int_{9}^{\infty}\frac{1}{\sqrt{x^3+1}}\,dx$$
The answer is that it converges, but why? I am so confused about this kind of questions. I tried to use the comparison test, so that $$\int_{9}^{\infty}\frac{1}{\sqrt{x^3+1}}\,dx<\int_{9}^{\infty}\frac{1}{\sqrt{x^3}}\,dx<\int_{9}^{\infty}\frac{1}{x}\,dx$$
but $\int_{9}^{\infty}\frac{1}{x}\,dx$ diverges. The thing is, the divergence of one integral doesn't tell us anything about the smaller integral. I have no idea how to solve this kind of questions. Can anyone explain to me how to determine convergence/divergence in general?
Best Answer
$$\int_9^{\infty} \frac{dx}{x^{3/2} }$$ converges, since it is $p$-integral with $p> 1$. Hence, by the comparison test, your integral must converge!