## Calculus (3rd Edition)

The integral $\int_{5}^{\infty} \dfrac{dx}{x^p \ln x}$ converges.
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.