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