Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 461: 90



Work Step by Step

Given $$\int_{0}^{\infty} e^{-x^{3}} d x$$ Since for $x\geq 1$, \begin{align*} e^x&\geq x\\ e^{x^3}&\geq x^3\\ e^{-x^3}&\leq x^{-3} \end{align*} and $\int_{1}^{\infty } x^{-3}dx$ converges. Since \begin{align*} \int_{0}^{\infty} e^{-x^{3}} d x&=\int_{0}^{1} e^{-x^{3}} d x+\int_{1}^{\infty} e^{-x^{3}} d x \end{align*} and $\int_{0}^{1} e^{-x^{3}} d x$ is finite, hence $\int_{0}^{\infty} e^{-x^{3}} d x$ converges.
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