Let $I = \int {\frac{{\sin x + \cos x}}{{\sqrt {1 + \sin 2x} }}} dx = \int {\frac{{(\sin x + \cos x)}}{{\sqrt {{{\sin }^2}x + {{\cos }^2}x + 2\sin x\cos x} }}} dx$


$ = \int {\frac{{\sin x + \cos x}}{{\sqrt {{{(\sin x + \cos x)}^2}} }}} dx = \int 1 dx = x + C$

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