# Integrate: (sin x -cos x) / 1+sin 2x

Solution:

$I=\int \frac{(\sin x -\cos x)}{1+\sin 2x}dx \\ \\ \Rightarrow \hspace{1cm} I=\int \frac{(\sin x -\cos x)}{\sin^{2}x+\cos^{2}x+2\sin x \cdot \cos x}dx \\ \\ \Rightarrow \hspace{1cm}I=\int \frac{(\sin x-\cos x)}{(\sin x-\cos x)^{2}}dx \hspace{0.5cm}Let \hspace{0.5cm}\sin x+\cos x=t \\ \\ \Rightarrow \hspace{1cm}(\cos x-\sin x)dx=dt \\ \\ \Rightarrow \hspace{1cm}I=-\int \frac{1}{t^{2}}dt=\frac{1}{t}+c=\frac{1}{\sin x+\cos x}+c$

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