Defining the function $h:(1,\infty)\to\mathbb{R}$ by $$h(x)=\int_1^x \ln(t^4 + 1)\mbox{ }dt,$$ I wish to find the smallest integer $k$ such that $h(x) = o(x^k) \mbox{ as } x\to\infty.$
That is, for all $\varepsilon>0$ there exists $x_0$ such that $|h(x)| < \varepsilon |x^k|$ whenever $x>x_0$. Since $h$ is only defined for $x>1$, both sides will be positive, and so we can drop the absolute value signs.
My intuition tells me that the answer is $k=5$, because $\ln(t^4 + 1)=o(t^4)$ as $t\to\infty$. However, I'm not sure how to give a rigorous proof using the formal definition, nor that this is indeed the smallest such $k$.