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: 47


The integral converges and is equal to: $0$

Work Step by Step

First, we will compute the integral for the negative interval $\int_{-\infty}^{\infty} xe^{-x^2} \ dx= \int_{-\infty}^0 xe^{x^2} \ dx \\=\lim\limits_{R \to \infty} \int_{R}^0 x e^{-x^2} \ dx \\=\lim\limits_{R \to \infty} (-\dfrac{1}{2}e^{-R^2}) \\=0$ Next, we will compute the integral for the positive interval $\int_{0}^{\infty} e^{-x^2} \ dx= \lim\limits_{R \to \infty}\int_{0}^{\infty} x e^{-x^2} \ dx \\=\lim\limits_{R \to \infty} (-\dfrac{1}{2}e^{-R^2}) \\=0$ Hence, the integral converges and yields the result: $\int_{-\infty}^{\infty} xe^{-x^2} \ dx=0+0=0$
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