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