Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.1 Basic Approaches - 7.1 Exercises: 50

Answer

\[ = 3\]

Work Step by Step

\[\begin{gathered} \int\limits_0^{\frac{\pi }{4}} {3\sqrt {1 + \sin 2x} } \,\,dx \hfill \\ \hfill \\ {\text{using}}\,\,{\text{the identity}}\,\,\,\,1 = {\sin ^2}x + {\cos ^2}x\,\, \hfill \\ and\,\,\,\sin \,\,\,2x = \,2\sin x\cos x \hfill \\ \hfill \\ = 3\int\limits_0^{\frac{\pi }{4}} {\sqrt {{{\sin }^2}x + {{\cos }^2}x + 2\sin x\cos x} \,dx} \hfill \\ \hfill \\ simplify\,\,inside\,\,the\,\,radical \hfill \\ \hfill \\ = 3\int\limits_0^{\frac{\pi }{4}} {\sqrt {\,{{\left( {\sin x + \cos x} \right)}^2}} } \,dx \hfill \\ \hfill \\ = 3\int\limits_0^{\frac{\pi }{4}} {\,\left( {\sin x + \cos x} \right)\,\,dx} \hfill \\ \hfill \\ \,integrate\,and\,use\,\,the\,\,ftc \hfill \\ \hfill \\ = 3\,\,\left[ { - \cos x + \sin x} \right]_0^{\frac{\pi }{4}} \hfill \\ \hfill \\ = 3\,\,\left[ { - \frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }} - \,\left( { - 1 + 0} \right)} \right] = 3 \hfill \\ \end{gathered} \]
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