Question:
Mark the correct alternative in each of the following:
The value of $\int \frac{\sin x+\cos x}{\sqrt{1-\sin 2 x}} d x$ is equal to
A. $\sqrt{\sin 2 x}+C$
B. $\sqrt{\cos 2 x}+C$
C. $\pm(\sin x-\cos x)+C$
D. $\pm \log (\sin x-\cos x)+C$
Solution:
$I=\int \frac{\sin x+\cos x}{\sin x-\cos x} d x(\sqrt{1-\sin 2 x}=\pm\{\sin x-\cos x\})$
Let $\mathrm{t}=\sin x-\cos x\left(\frac{d t}{d x}=\sin x+\cos x\right)$
$I=\int \frac{d t}{t}$
$I=\pm \log (\sin x-\cos x)+c$
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