Calculus 8th Edition

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

Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 12


$\frac{tan(2\theta)}{2} +C$

Work Step by Step

Evaluate the Integral using substitution: $\int sec^2(2\theta)d\theta$ Substitution Rule: $\int f(g(x))gā€™(x)dx = \int f(u)du$ $u= 2\theta$ $du =2$ Since $du$ in the expression is equal to $1$ it must be multiplied by $\frac{1}{2} $ Solve the integral in terms of $u$: $\int sec^2(u)(\frac12)du$ $\frac{1}{2}\int sec^2(u)du $ $\frac{1}{2}tan(u) +C$ $\frac{tan(u)}{2} + C$ Substitute for $u$: $\frac{tan(2\theta)}{2} +C$
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