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: 92

Answer

$F^{\prime}(x)=2x\sin x^{4}$

Work Step by Step

See Example 8. Substituting $u=x^{2},\displaystyle \quad \frac{du}{dx}=2x$ $\displaystyle \frac{dF}{dx}=\frac{dF}{du}\cdot\frac{du}{dx}$ $=\displaystyle \frac{d}{du}[F(x)]\cdot\frac{du}{dx}$ $=\displaystyle \frac{d}{du}[\int_{0}^{x^{2}}\sin\theta^{2}d\theta]\cdot\frac{du}{dx}$ ... apply the substitution ... $=\displaystyle \frac{d}{du}[\int_{0}^{u}\sin\theta^{2}d\theta]\cdot 2x$ ... apply the 2nd FTC, $\displaystyle \frac{d}{dx}[\int_{a}^{x}f(t)dt]=f(x)$. $= \sin u^{2}\cdot 2x$ ... bring x back ... $= \sin(x^{2})^{2}\cdot 2x$ $F^{\prime}(x)=2x\sin x^{4}$
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