Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.1 Integration by Parts - Exercises - Page 395: 15


$$\int e^{-x}sin\,x\,dx=-\frac{e^{-x}}{2}(cos\,x+sin\,x)+C$$

Work Step by Step

$$\int e^{-x}sin\,x\,dx=-e^{-x}cos\,x-\int e^{-x}cos\,x\,dx$$ $$=-e^{-x}cos\,x-(e^{-x}sin\,x+\int e^{-x}sin\,x\,dx)$$ $$2\int e^{-x}sin\,x\,dx=-e^{-x}cos\,x-e^{-x}sin\,x$$ $$\int e^{-x}sin\,x\,dx=-\frac{e^{-x}}{2}(cos\,x+sin\,x)+C$$
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