I'm currently trying to prove the existence of the following improper integral:
$$I := \int_0^\infty \frac{\cos(x)}{\sqrt{x}+x^3} dx$$
and to show that $|I| \leq \frac{5}{2}$.
$\begin{align}|I| &\leq \int_0^\infty \frac{|\cos(x)|}{|\sqrt{x}+x^3|} dx\\ &\leq \int_0^\infty \frac{1}{\sqrt{x}+x^3} dx\end{align}$
However, I'm pretty much stuck at this point, I could use a substitution $u = \sqrt{x}$ which leads to $$ 2 \cdot \int_0^\infty \frac{1}{1+u^5}du$$ but how can I evaluate this improper integral the easiest way to get the upper bound?