Calculus: Early Transcendentals 8th Edition

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

Chapter 7 - Section 7.5 - Strategy for Integration - 7.5 Exercises: 7

Answer

$$\displaystyle\int_{-1}^{1}\frac{e^{\arctan{y}}}{1+y^{2}}\thinspace dy=e^{\frac{\pi}{4}}-e^{\frac{-\pi}{4}}$$

Work Step by Step

$$\displaystyle\int_{-1}^{1}\frac{e^{\arctan{y}}}{1+y^{2}}\thinspace dy$$ $u= \arctan{y}\quad du=\frac{dy}{1+y^{2}}$ Limits of integration will change from $\displaystyle\int_{-1}^{1}$ to $\displaystyle\int_{\arctan{(-1)}}^{\arctan{(1)}} = \displaystyle\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}$ $$ \displaystyle\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}e^{u}\thinspace du$$ $$=\bigg[e^{u}\bigg]^{\frac{\pi}{4}}_{\frac{-\pi}{4}}$$ $$=e^{\frac{\pi}{4}}-e^{\frac{-\pi}{4}}$$
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