Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.2 Calculus of Vector-Valued Functions - Exercises - Page 720: 40


$\left\langle \frac{\pi}{4}, \frac{1}{2}\ln(2) \right\rangle$

Work Step by Step

We have \begin{align} \int_{0}^{1}\left\langle \frac{1}{1+s^2},\frac{s}{1+s^2}\right\rangle ds&=\left\langle \tan^{-1}s, \frac{1}{2}\ln(s^2+1)\right\rangle|_{0}^{1}\\ &=\left\langle \tan^{-1}1, \frac{1}{2}\ln(2)\right\rangle-\left\langle \tan^{-1}0, \frac{1}{2}\ln(1)\right\rangle\\ &= \left\langle \frac{\pi}{4}, \frac{1}{2}\ln(2) \right\rangle. \end{align}
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