Answer
The integral $\int_{5}^{\infty} \dfrac{dx}{x^2 \ln x}$ converges as well.
Work Step by Step
We are given the function
$f(x)=\int_{5}^{\infty} \dfrac{dx}{x^2 \ln x}$
Since, $x \gt 5$ and $x^2 \gt 0$
$\dfrac{1}{x^2 \ln x} \lt \dfrac{1}{x^2} $
But the integral $\int_{5}^{\infty} \dfrac{dx}{x^2} $ shows a p-type integral with $2 \gt 1$. Thus, the integral $\int_{5}^{\infty} \dfrac{dx}{x^2} $ converges.
Therefore, by the comparison test, the integral $\int_{5}^{\infty} \dfrac{dx}{x^2 \ln x}$ converges as well.
