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 440: 26



Work Step by Step

\begin{aligned} \int_{-\infty}^{0} x e^{-x^{2}} d x &=\lim _{R \rightarrow-\infty} \int_{R}^{0} x e^{-x^{2}} d x \\ &=\lim _{R \rightarrow-\infty} -\frac{1}{2} \int_{R}^{0}(-2 x) e^{-x^{2}} d x \\ &=\lim _{R \rightarrow-\infty}-\frac{1}{2} e^{-(x)^{2}} \bigg|_{R}^{0} \\ &=\lim _{R \rightarrow-\infty}\frac{1}{2} e^{-R^{2}}-\frac{1}{2}\\ &= \frac{-1}{2} \end{aligned} Converges.
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