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 290: 91


$F^{\prime}(x)=3x^{2}\sin x^{6}$

Work Step by Step

See Example 8. Substituting $u=x^{3},\displaystyle \quad \frac{du}{dx}=3x^{2}$ Chain Rule: $\displaystyle \frac{dF}{dx}=\frac{dF}{du}\cdot\frac{du}{dx}$ $=\displaystyle \frac{d}{du}[F(x)]\cdot\frac{du}{dx}$ $=\displaystyle \frac{d}{du}[\int_{0}^{x^{3}}\sin t^{2}dt]\cdot\frac{du}{dx}$ ... apply the substitution ... $=\displaystyle \frac{d}{du}[\int_{0}^{u}\sin t^{2}dt]\cdot 3x^{2}$ ... apply the 2nd FTC, $= \sin u^{2}\cdot 3x^{2}$ ... bring x back ... $= \sin(x^{3})^{2}\cdot 3x^{2}$ $F^{\prime}(x)=3x^{2}\sin x^{6}$
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