Write a value

We know that

$1+\sin 2 x=\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=(\sin x+\cos x)^{2}$

$y=\int \frac{\sin x+\cos x}{\sqrt{(\sin x+\cos x)^{2}}} d x$

$y=\int \frac{(\sin x+\cos x)}{(\sin x+\cos x)} d x$

$y=\int d x$

Use formula $\int c d x=c x$, where $c$ is constant

$y=x+c$