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: 57


Equation of the tangent: $x+\pi y-\pi=0$.

Work Step by Step

$F(x)=\int_\pi^x\frac{\cos t}{t}dt$. Differentiating with respect to $x$, keeping fundamental theorem of calculus-I in our mind, $$F'(x)=\frac{\cos x}{x}$$Thus, the slope of the tangent line at $x=\pi$, $$F'(\pi)=\frac{\cos \pi}{\pi}=-\frac{1}{\pi}$$Using the point slope form, $$y-F(\pi)=-\frac{1}{\pi}(x-\pi)$$Since $F(\pi)=0$,$$\pi y=\pi-x\implies x+\pi y-\pi=0.$$
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