Prove the following
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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