Find:
     \int \frac{\sin x-\cos x}{\sqrt{1+\sin 2x}}dx, \: 0< x< \frac{\pi }{2}

 

 

 

 
 
 
 
 

Answers (1)

Let I= \int \frac{\sin x-\cos x}{\sqrt{1+\sin 2x}}dx,0< x< \frac{\pi }{2}
     I= \int \frac{\sin x-\cos x}{\sqrt{\sin^{2}x+\cos ^{2}x+2\sin x\cos x}}dx
       = \int \frac{\sin x-\cos x}{\sqrt{\left ( \sin x+\cos x \right )^{2}}}dx\Rightarrow \int \frac{\sin x-\cos x}{\left ( \sin x+\cos x \right )}dx
\Rightarrow let \: \sin x+\cos x= t\Rightarrow \left ( \cos x-\sin x \right )dx= dt
I= \int \frac{-1}{t}dt= -\log \left | t \right |+C
\Rightarrow -\log \left | \cos x+\sin x \right |+C
\because m\log n= \log n^{m}\Rightarrow \log \left | \cos x+\sin x \right |^{-1}+C

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