Calculus: Early Transcendentals 8th Edition

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

Chapter 5 - Section 5.4 - Indefinite Integrals and the Net Change Theorem - 5.4 Exercises - Page 409: 27


$$\int^{\pi}_0(5e^x+3\sin x)dx=5e^\pi+1$$

Work Step by Step

$$A=\int^{\pi}_0(5e^x+3\sin x)dx$$ According to Table 1, we have $$\int cf(x)dx=c\int f(x)dx$$ $$\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$$ Therefore, $$A=5\int^{\pi}_0(e^x)dx+3\int^{\pi}_0(\sin x)dx$$ Also, from Table 1, $$\int (e^x)dx=e^x$$ $$\int(\sin x)dx=-\cos x$$ Therefore, $$A=5(e^x)\Bigg]^\pi_0-3\cos x\Bigg]^\pi_0$$ $$A=5(e^\pi-e^0)-3(\cos\pi-\cos0)$$ $$A=5(e^\pi-1)-3(-1-1)$$ $$A=5e^\pi-5-3\times(-2)$$ $$A=5e^\pi+1$$
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