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: 53

Answer

$$3 \tan ^{4} x-2 \tan ^{3} x+C$$

Work Step by Step

Given $$\int \sec ^{2}(x) \cdot\left(12 \tan ^{3} x-6 \tan ^{2} x\right) d x $$ Let $$ u=\tan x \ \ \ \ \Rightarrow \ \ \ du = \sec^2 x dx$$ Then \begin{aligned} \int \sec ^{2}(x) \cdot\left(12 \tan ^{3} x-6 \tan ^{2} x\right) d x &=\int\left(12 u^{3}-6 u^{2}\right) d u \\ &=3 u^{4}-2 u^{3}+C \\ &=3 \tan ^{4} x-2 \tan ^{3} x+C \end{aligned}
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