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: 8

Answer

$$\displaystyle\int t\sin{t}\cos{t}\thinspace dt=\frac{\sin{2t}}{8}-\frac{t\cos{2t}}{4}+C$$

Work Step by Step

Use the half angle trignometric identity $\sin{2\theta}=2\sin{\theta}\cos{\theta}$ to rewrite $\sin{\theta}\cos{\theta}=\frac{\sin{2\theta}}{2}$ $\displaystyle\int t\sin{t}\cos{t}\thinspace dt = \displaystyle\int t\frac{\sin{2t}}{2}\thinspace dt$ Pull the constant outside $\frac{1}{2}\int t\sin{2t}\thinspace dt$ Integration by parts: $uv-\int v\thinspace du$, let $u =t$ and $dv=\sin(2t)dt$ then, $du=dt$ and $v=\big(\frac{1}{2}\big)(-\cos{2t})$ $$=\frac{1}{2}\bigg[t\bigg(\frac{1}{2}\bigg)(-\cos{2t})-\int\bigg(\frac{1}{2}\bigg)(-\cos{2t})\thinspace dt\bigg]$$ $$=\frac{1}{2}\bigg[t\bigg(\frac{1}{2}\bigg)(-\cos{2t})+\int\bigg(\frac{1}{2}\bigg)(\cos{2t})\thinspace dt\bigg]$$ $$=\frac{1}{2}\Bigg[t\bigg(\frac{1}{2}(-cos{2t})+\bigg(\frac{1}{2}\bigg)\bigg(\frac{1}{2}\bigg)\sin{2t}+C\Bigg]$$ $$=\frac{1}{2}\Bigg[\bigg(\frac{-t\cos{2t}}{2}\bigg)+\bigg(\frac{\sin{2t}}{4}\bigg)+C\Bigg]$$ $$=\bigg(\frac{-tcos{2t}}{4}\bigg)+\bigg(\frac{\sin{2t}}{8}\bigg)+C$$ $$=\frac{\sin{2t}}{8}-\frac{t\cos{2t}}{4}+C$$
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