Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.7 Improper Integrals - Exercises - Page 441: 71


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.
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