Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 7 - Section 7.1 - Integration by Parts - 7.1 Exercises - Page 476: 4


$$\int ye^{0.2y}dy=5ye^{0.2y}-25e^{0.2y}+C$$

Work Step by Step

$$A=\int ye^{0.2y}dy$$ We would choose $u=y$ and $dv=e^{0.2y}dy$ We could easily see that $du=dy$. For $dv=e^{0.2y}dy$, we analyze as follows $$\int e^{0.2y}dy=\int\frac{1}{0.2}e^{0.2y}d(0.2y)=5\int e^{0.2y}d(0.2y)=5e^{0.2y}+C$$ Therefore, for $dv=e^{0.2y}dy$, $v=5e^{0.2y}$ Apply Integration by Parts to A, we have $$A=uv-\int vdu$$ $$A=5ye^{0.2y}-\int5e^{0.2y}dy$$ $$A=5ye^{0.2y}-5\int e^{0.2y}dy$$ $$A=5ye^{0.2y}-5(5e^{0.2y}+C)$$ (as we have analyzed above) $$A=5ye^{0.2y}-25e^{0.2y}+C$$
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