Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.8 - Improper Integrals - Exercises 8.8 - Page 501: 22



Work Step by Step

Use the formula $$\int e^{a x} \sin b x d x=\frac{e^{a x}}{a^{2}+b^{2}}(a \sin b x-b \cos b x)+C$$ We get \begin{align*} \int_{0}^{\infty} 2 e^{-\theta} \sin \theta d \theta&=\lim _{b \rightarrow \infty} \int_{0}^{b} 2 e^{-\theta} \sin \theta d \theta\\ &=\lim _{b \rightarrow \infty} 2\left[\frac{e^{-\theta}}{1+1}(-\sin \theta-\cos \theta)\right]_{0}^{b}\\ &=\lim _{b \rightarrow \infty}\left[\frac{-2(\sin b+\cos b)}{2 e^{b}}+\frac{2(\sin 0+\cos 0)}{2 e^{0}}\right]\\ &=0+\frac{2(0+1)}{2}\\ &=1 \end{align*}
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