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: 2

Answer

Verify $\int\cos^2{x}dx=\frac{1}{2}x+\frac{1}{4}\sin{2x}+C$ We need to take the derivative of the right side of our equation and verify that it equals the expression inside of our integral. \begin{equation*} \frac{d}{dx}\left(\frac{1}{2}x+\frac{1}{4}\sin{2x}+C\right)=\frac{d}{dx}\left(\frac{1}{2}x\right)+\frac{d}{dx}\left(\frac{1}{4}\sin{2x}\right)+\frac{d}{dx}\left(C\right) \end{equation*} \begin{equation*} =\left(\frac{1}{2}\right)+\left(\frac{1}{4}\cos{2x}\times2\right)+\left(0\right) \end{equation*} \begin{equation*} =\frac{1}{2}+\frac{1}{2}\cos{2x}=\frac{1+\cos{2x}}{2}=\cos^2{x} \end{equation*} as desired.

Work Step by Step

Verify $\int\cos^2{x}dx=\frac{1}{2}x+\frac{1}{4}\sin{2x}+C$ We need to take the derivative of the right side of our equation and verify that it equals the expression inside of our integral. \begin{equation*} \frac{d}{dx}\left(\frac{1}{2}x+\frac{1}{4}\sin{2x}+C\right)=\frac{d}{dx}\left(\frac{1}{2}x\right)+\frac{d}{dx}\left(\frac{1}{4}\sin{2x}\right)+\frac{d}{dx}\left(C\right) \end{equation*} \begin{equation*} =\left(\frac{1}{2}\right)+\left(\frac{1}{4}\cos{2x}\times2\right)+\left(0\right) \end{equation*} \begin{equation*} =\frac{1}{2}+\frac{1}{2}\cos{2x}=\frac{1+\cos{2x}}{2}=\cos^2{x} \end{equation*} as desired.
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