Calculus 8th Edition

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

Chapter 4 - Integrals - 4.3 The Fundamental Theorem of Calculus - 4.3 Exercises - Page 328: 51


$f(\theta)=\sec\theta\tan\theta$ is not continuous in $(\pi/3,\pi)$, and hence FTC-2 cannot be applied.

Work Step by Step

According to the fundamental theorem of calculus, $\int_a^bf(x)dx = F(b)-F(a)$, where $F$ is the antiderivative of $f$ or $F'=f$, provided that $f$ is continuous in $(a, b)$. Here, $\sec\theta\tan\theta$ is not continuous in $(\pi/3,\pi)$ and has an infinite discontinuity at $\theta=\pi/2$.
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