Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 35


$$\int_0^1\cos(\pi t/2)dt=\frac{2}{\pi}$$

Work Step by Step

To evaluate the integral $$\int_0^1\cos(\pi t/2)dt$$ we will use substitution $\pi t/2=z$ which gives us $\pi/2dt=dz\Rightarrow dt=2/\pi dx$ and now integration bounds would be: for $t=0$ we have $z=0$ and for $t=1$ we have $z=\pi/2.$ Now, putting all this into the integral we get: $$\int_0^1\cos(\pi t/2)dt=\int_0^{\pi/2}\cos z\cdot2/\pi dz= \frac{2}{\pi}\left.\sin z\right|_0^{\pi/2}=\frac{2}{\pi}(\sin\frac{\pi}{2}-\sin0)=\frac{2}{\pi}(1-0)=\frac{2}{\pi}$$
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