Calculus: Early Transcendentals 8th Edition

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

Chapter 5 - Section 5.3 - The Fundamental Theorem of Calculus - 5.3 Exercises: 37

Answer

$\frac{e^2}{e+1}$

Work Step by Step

$\int^1_0 (x^e + e^x)dx$ The first step is to separate into 2 definite integrals: $ = \int^1_0 x^e dx + \int^1_0 e^x dx$ Solve both definite integrals: $ = (\frac{x^{e+1}}{e+1})|^1_0 + (e^x)|^1_0$ Plug in the limits of integration: $ = [\frac{(1)^{e+1}}{e+1} - \frac{(0)^{e+1}}{e+1}] + [e^1 - e^0]$ Simplify until final answer is reached (remember that $1^{e+1} = 1$): $= [\frac{1}{e+1}- 0] + [e - 1]$ $= \frac{1}{e+1} + e - 1$ $= \frac{1}{e+1} + \frac{e^2 + e}{e+1} - \frac{e+1}{e+1}$ $= \frac{e^2}{e+1}$
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