Calculus 8th Edition

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

Chapter 4 - Integrals - Problems Plus - Problems - Page 353: 1

Answer

$$ f(4)=\frac{\pi}{2} $$

Work Step by Step

$$ x \sin \pi x=\int_{0}^{x^{2}} f(t) d t $$ Differentiating both sides of this equation $$ \begin{aligned} \frac{d}{dx} \left[ x \sin \pi x \right] &= \frac{d}{dx} \left[\int_{0}^{x^{2}} f(t) d t \right]\\ \sin \pi x+\pi x \cos \pi x &= \frac{d}{dx} \left[\int_{0}^{x^{2}} f(t) d t \right]\\ \end{aligned} $$ using $\mathrm{FTC} 1$ and the Chain Rule for the right side) gives $$ \begin{aligned} \sin \pi x+\pi x \cos \pi x &= 2 x f\left(x^{2}\right) \end{aligned} $$ Letting $x=2$ so that $f\left(x^{2}\right)=f(4),$ we obtain $$\sin 2 \pi+2 \pi \cos 2 \pi=4 f(4),$$ so $$f(4)=\frac{1}{4}(0+2 \pi \cdot 1)=\frac{\pi}{2}$$
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