Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 5 - The Integral - 5.7 Substitution Method - Exercises - Page 276: 54

Answer

$$\frac{5}{4} \sec \left(x^{4 / 5}\right)+c$$

Work Step by Step

Given $$\int x^{-1 / 5} \sec \left(x^{4 / 5}\right) \tan \left(x^{4 / 5}\right) dx$$ Let $$ u=x^{4 / 5} \ \ \ \ \Rightarrow \ \ \ du =\frac{5}{4} x^{-1/5}dx$$ Then \begin{aligned} \int x^{-1 / 5} \sec \left(x^{4 / 5}\right) \tan \left(x^{4 / 5}\right) dx&=\frac{5}{4} \int \sec (u) \tan (u) d u \\ &=\frac{5}{4} \sec u +c\\ &=\frac{5}{4} \sec \left(x^{4 / 5}\right)+c\end{aligned}
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