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 442: 77

Answer

The integral $\int_{0}^{\infty} \dfrac{dx}{(x^{1/2}(x+1))}$ converges.

Work Step by Step

We are given the function $f(x)=\int_{0}^{\infty} \dfrac{dx}{(x^{1/2}(x+1))}$ Since, $1 \leq 1+x$ This yields: $\dfrac{dx}{x^{1/2}(x+1)} \leq \dfrac{1}{x^{1/2}} $ Consider the integral $\int_{0}^{1} \dfrac{dx}{x^{1/2}}=[2x^{1/2}]_0^1 \\=2-0\\=2$ Thus, the integral $\int_{0}^{1} \dfrac{dx}{x^{1/2}}$ converges. Therefore, by the comparison test, the integral $\int_{0}^{\infty} \dfrac{dx}{(x^{1/2}(x+1))}$ converges as well.
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