Answer
converges
Work Step by Step
Given $$\int_{3}^{\infty} \frac{d x}{x^{4}+\cos ^{2} x} $$
Since for $x\geq 1$
$$\frac{1}{x^{4}+\cos ^{2} x} \geq \frac{1}{x^{4}} $$
and
\begin{align*}
\int_{1}^{\infty} \frac{d x}{x^{4} }&= \lim_{R\to \infty } \int_{1}^{R} \frac{d x}{x^{4} }\\
&= \lim_{R\to \infty } -\frac{1}{3x^3}\bigg|_{1}^{R} \\
&=\frac{1}{3}
\end{align*}
and $x^{4}+\cos ^{2} x\neq 0$, so $ \int_{1}^{3} \frac{d x}{x^{4}+\cos ^{2} x}$ is finite. Hence, $ \int_{3}^{\infty} \frac{d x}{x^{4}+\cos ^{2} x}$ converges
