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: 8


$$3(\ln 3-1) \mathbf{i}+(3-e) \mathbf{j}+(\ln 3(\ln (\ln 3)-1)+1) \mathbf{k} $$

Work Step by Step

We evaluate the integral of the vector function as follows: \begin{align*} \int_{1}^{\ln 3}\left(t e^{t} \mathbf{i}+e^{t} \mathbf{j}+\ln t \mathbf{k}\right) d t&= \left( te^t\bigg|_{1}^{\ln 3} - \int_{1}^{\ln 3} e^tdt\right)\mathbf{i}-\left[e^{t}\right]_{1}^{\ln 3} \mathbf{j}+[t \ln t-t]_{1}^{\ln 3} \mathbf{k} \\ &=\left[t e^{t}-e^{t}\right]\bigg|_{1}^{\ln 3} \mathbf{i}-\left[e^{t}\right]\bigg|_{1}^{\ln 3} \mathbf{j}+[t \ln t-t]\bigg|_{1}^{\ln 3} \mathbf{k} \\ &=3(\ln 3-1) \mathbf{i}+(3-e) \mathbf{j}+(\ln 3(\ln (\ln 3)-1)+1) \mathbf{k} \end{align*}
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