Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 4 - Integration - 4.4 Exercises - Page 291: 108

Answer

The Fundamental Theorem of Calculus does not apply (for step 1).

Work Step by Step

Th.4.9 The Fundamental Theorem of Calculus: If a function $f$ is CONTINUOUS on the closed interval $[a, b]$ and $F$ is an antiderivative of $f$ on the interval $[a, b]$, then $\displaystyle \int_{\mathit{0}}^{b}f(x)dx=F(b)-F(a)$. ------------- We may apply the FTC when $f(x)=\csc x\cot x $ is continuous on the closed interval $[a, b]$ $f(x)=\csc x\cot x$ is not continuous on $[\displaystyle \frac{\pi}{2}, \displaystyle \frac{3\pi}{2}]..$ It has a nonremovable discontinuity at $ x=\pi$. So, the first step is not allowed, as the theorem does not apply here.
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