Answer
The integral $\int_{1}^{\infty} \dfrac{dx}{x^2}$ converges to $1$.
Hence, the integral $\int_1^\infty \dfrac{dx}{\sqrt {1+x^4}}$ converges as well by the comparison test.
Work Step by Step
Since, $\sqrt {x^4+1} \geq \sqrt {x^4}(=x^2)$, it follows that$\dfrac{1}{\sqrt {x^4+1}}\leq \dfrac{1}{x^2}$
We will compute the integral as follows:
$\int_{1}^{\infty} \dfrac{dx}{x^2}= \lim\limits_{R \to \infty} \int_{1}^R \dfrac{dx}{x^2} \\=\lim\limits_{R \to \infty} [-\dfrac{1}{x}]_1^R\\=\lim\limits_{R \to \infty}(-\dfrac{1}{R}+1)\\=1$
So, the integral $\int_{1}^{\infty} \dfrac{dx}{x^2}$ converges to $1$.
Hence, the integral $\int_1^\infty \dfrac{dx}{\sqrt {1+x^4}}$ converges as well by the comparison test.
