Question:
$\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2 x}} d x$
Solution:
Let $\mathrm{I}=\int \frac{\sin x+\cos x}{\sqrt{1+2 \sin x \cos x}} d x$
$=\int \frac{(\sin x+\cos x)}{\sqrt{\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x}} d x$
$=\int \frac{\sin x+\cos x}{\sqrt{(\sin x+\cos x)^{2}}} d x=\int \frac{\sin x+\cos x}{\sin x+\cos x} d x$
$=\int 1 d x$
$=\int 1 d x$
$=x+C$ , where $C$ is a constant
Therefore,
$\mathrm{I}=\mathrm{x}+\mathrm{C}$
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