Question:
Write a value of $\int \frac{\sin x-\cos x}{\sqrt{1+\sin 2 x}} d x$
Solution:
We know that
$1+\sin 2 x=\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x$
$=(\sin x+\cos x)^{2}$
$y=\int \frac{\sin x-\cos x}{\sqrt{(\sin x+\cos x)^{2}}} d x$
$y=\int \frac{(\sin x-\cos x)}{(\sin x+\cos x)} d x$
Let, $\sin x+\cos x=t$
Differentiating both sides with respect to $x$
$\frac{d t}{d x}=\cos x-\sin x$
$\Rightarrow-\mathrm{dt}=(\sin x-\cos x) d x$
$y=\int \frac{-1}{t} d t$
Use formula $\int \frac{1}{t}=\log t$
$y=-\log t+c$
Again, put $t=\sin x+\cos x$
$y=-\log (\sin x+\cos x)+c$