Calculus: Early Transcendentals 8th Edition

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

Chapter 5 - Section 5.5 - The Substitution Rule - 5.5 Exercises: 53

Answer

$$\frac{2}{\pi}$$

Work Step by Step

By substitution: Let $u = \frac{\pi}{2}t$. Hence, $du = \frac{\pi}{2}dt$. Thus, substituting $\frac{\pi}{2}t$ with $u$ and $dt$ with $\frac{2}{\pi}du$: $\int^1_0\cos({\frac{\pi}{2}t})dt$ $=\int^{\frac{\pi}{2}}_0\cos u (\frac{2}{\pi}du)$ $=\frac{2}{\pi}\int^{\frac{\pi}{2}}_0\cos u du$ $=\frac{2}{\pi}[\sin u]|^{\frac{\pi}{2}}_0$ $=\frac{2}{\pi}[\sin(\frac{\pi}{2})-\sin 0]$ $=\frac{2}{\pi}[1-0]$ $=\frac{2}{\pi}$
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