Calculus 8th Edition

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

Chapter 4 - Integrals - 4.4 Indefinite Integrals and the Net Change Theorem - 4.4 Exercises: 16

Answer

$2sin(x)+c$

Work Step by Step

Given $\int \dfrac{sin(2x)}{sin(x)}dx$ 1.) Using the double angle identitie: $sin(2x)=2sin(x)cos(x) , $ we have: $\int \dfrac{sin(2x)}{sin(x)}dx = \int \dfrac{2sin(x)cos(x)}{sin(x)}dx \\$ 2.) Simplifyng: $\int \dfrac{2sin(x)cos(x)}{sin(x)}dx = \int 2cos(x)dx \\$ 3.) Taking out the constant multiple: $ \int 2cos(x)dx = 2\int cos(x)dx \\$ 4.) Solving the integral: $ 2\int cos(x)dx= 2(sin(x)+c) \\$ 5.) Performing the distributive property: $2(sin(x)+c)= 2sin(x)+2c \\$ 6.) Since a constant ($c$) times $2$ it's still a constant: $2sin(x)+2c= 2sin(x)+c$
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