Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.2 - Integrals of Vector Functions; Projectile Motion - Exercises 13.2 - Page 753: 4


$$\mathbf{i}+(\ln 2) \mathbf{j}+\frac{3}{4} \mathbf{k} $$

Work Step by Step

Given $$ \int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t $$ Then \begin{align*} \int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t&=\int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(\sin 2 t) \mathbf{k}] d t\\ &=[\sec t]\bigg|_{0}^{\pi / 3} \mathbf{i}+[-\ln (\cos t)]\bigg|_{0}^{\pi / 3} \mathbf{j}+\left[-\frac{1}{2} \cos 2 t\right]\bigg|_{0}^{\pi / 3} \mathbf{k}\\ &=\mathbf{i}+(\ln 2) \mathbf{j}+\frac{3}{4} \mathbf{k} \end{align*}
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